Series Tower

Floor 1

Hall of Beginnings

Your journey with series begins here. Do you have the strength to add up infinitely many numbers?

\[ \sum_{n=1}^\infty \left(\frac{1}{2}\right)^n \]

First Few Terms: \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots\)
Convergence Type: Convergent
Sum: 1
Description: The most infamous geometric series, related to Zeno’s dichotomy paradox. As the pieces get smaller and smaller, the whole unit is completed.
An infinite number of mathematicians walk into a bar. The first orders half a beer, the second orders a quarter, the third orders an eighth, and so on. The bartender hands all of them a beer and says, “You guys need to know your limits.”

\[ \sum_{n=1}^\infty n \]

First Few Terms: \(1 + 2 + 3 + 4 + \cdots\)
Convergence Type: Divergent
Sum: \(\infty\)
Description: The numbers being added get larger and larger. Because of this, the partial sum becomes infinitely large, causing the series to diverge.
This series is like an ever-growing snowball.

\[ \sum_{n=1}^\infty (-1)^{n+1} \]

First Few Terms: \(1 - 1 + 1 - 1 + \cdots\)
Convergence Type: Divergent
Sum: Does Not Exist
Description: Also known as Grandi’s series, this series seems paradoxical. Is its sum 1, 0, or 1/2? The limit definition of a series refuses to give any sum at all, as the partial sums oscillate between 1 and 0 forever.
Schrödinger’s infinite series.

Floor 2

Geometric Series

You encounter series with a distinctive pattern of common ratios. Using the arts of algebra and geometry, can you find their sums?

\[ \sum_{n=0}^\infty \left(\frac{1}{3}\right)^n \]

First Few Terms: \(1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots\)
Convergence Type: Convergent
Sum: \(3/2 = 1.5\)
Description: A geometric series with a common ratio that isn’t 1/2.

\[ \sum_{n=0}^\infty \left(\frac{2}{3}\right)^n \]

First Few Terms: \(1 + \frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \cdots\)
Convergence Type: Convergent
Sum: 3
Description: A geometric series with a common ratio whose numerator in reduced form is not 1.
Slow but steady exponential decay leads to convergence.

\[ \sum_{n=0}^\infty \left(-\frac{1}{2}\right)^n \]

First Few Terms: \(1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots\)
Convergence Type: Convergent
Sum: 2/3
Description: A geometric series with a negative common ratio.

\[ \sum_{n=0}^\infty \left(\frac{3}{2}\right)^n \]

First Few Terms: \(1 + \frac{3}{2} + \frac{9}{4} + \frac{27}{8} + \cdots\)
Convergence Type: Divergent
Sum: \(\infty\)
Description: A geometric series with a common ratio greater than 1.
The power of uncontrolled exponential growth.

\[ \sum_{n=0}^\infty \left(-\frac{3}{2}\right)^n \]

First Few Terms: \(1 - \frac{3}{2} + \frac{9}{4} - \frac{27}{8} + \cdots\)
Convergence Type: Divergent
Sum: Does Not Exist
Description: A geometric series with a common ratio less than -1.

\[ \sum_{n=0}^\infty 3^n(2^{-2n}) \]

First Few Terms: \(1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots\)
Convergence Type: Convergent
Sum: 4
Description: A geometric series that requires manipulation to get into standard form.
Not all geometric series look alike.

Floor 3

Divergence Test

Some series refuse to converge to a finite sum, becoming uncontrollably large or oscillating forever. How can we identify these types of series?

\[ \sum_{n=1}^\infty \ln(n) \]

First Few Terms: \(\ln(1) + \ln(2) + \ln(3) + \ln(4) + \cdots\)
Convergence Type: Divergent
Sum: \(\infty\)
Description: A series which is divergent because \(\lim_{n\to\infty}a_n\) is infinite.
With endlessly growing terms, there’s no way this sum can be anything but infinite.

\[ \sum_{n=1}^\infty \sin(n) \]

First Few Terms: \(\sin(1) + \sin(2) + \sin(3) + \sin(4) + \cdots\)
Convergence Type: Divergent
Sum: Does Not Exist
Description: A series which is divergent because \(\lim_{n\to\infty}a_n\) doesn’t exist. This is due to \(\sin(x)\) oscillating forever as \(x\) approaches \(\infty\).
This series is like an infinitely oscillating wave.

\[ \sum_{n=1}^\infty \frac{2n^2 + 4n + 7}{n^2 - 2n - 17} \]

First Few Terms: \(- \frac{13}{18} - \frac{23}{17} - \frac{37}{14} - \frac{55}{9} - \cdots\)
Convergence Type: Divergent
Sum: \(\infty\)
Description: A divergence test example that requires finding a more complicated limit. \(\lim_{n\to\infty}a_n\) equals 2, so the series is divergent.
Don’t get distracted by all the polynomial terms!

\[ \sum_{n=1}^\infty \frac{e^n}{n} \]

First Few Terms: \(e + \frac{e^2}{2} + \frac{e^3}{3} + \cdots\)
Convergence Type: Divergent
Sum: \(\infty\)
Description: A divergence test example that requires finding a limit using L’Hôpital’s Rule.
Don’t forget your limit-finding techniques!

Floor 4

Telescoping Series

Sometimes, you just have to resort to using partial sums and limits. In doing so, you might find that many things disappear.

\[ \sum_{n=0}^\infty \frac{1}{(n+1)(n+2)} \]

First Few Terms: \(\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \cdots\)
Convergence Type: Convergent
Sum: 1
Description: A telescoping series where adjacent terms cancel.
Like closing a retractable telescope, most of the terms in the partial sums cancel.

\[ \sum_{n=0}^\infty \frac{2}{(n+1)(n+3)} \]

First Few Terms: \(\frac{2}{3} + \frac{2}{8} + \frac{2}{15} + \frac{2}{24} + \cdots\)
Convergence Type: Convergent
Sum: 3/2
Description: A telescoping series that requires cancelling non-adjacent terms.

\[ \sum_{n=2}^\infty \ln\left(\frac{n}{n+1}\right)\]

First Few Terms: \(\ln(\frac{2}{3}) + \ln(\frac{3}{4}) + \ln(\frac{4}{5}) + \ln(\frac{5}{6}) + \cdots\)
Convergence Type: Divergent
Sum: \(-\infty\)
Description: A telescoping series that seems to converge at first to \(\ln(2)\), but diverges when you take a closer look at its partial sums.
Telescoping series can be deceiving at first glance.

Floor 5

Integral Test / p-Series

Some of the secrets of series can be unlocked with the art of integration.

\[ \sum_{n=1}^\infty \frac{1}{n} \]

First Few Terms: \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots\)
Convergence Type: Divergent
Sum: \(\infty\)
Description: Also known as the harmonic series. Its divergence was first shown by Nicole Oresme in the 1300s with a proof that involves the reciprocals of powers of 2. Alternatively, you can use the integral test to show its divergence.
Not all series whose terms tend to 0 converge - some series have terms that decay too slowly.

\[ \sum_{n=1}^\infty \frac{1}{n^2} \]

First Few Terms: \(1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots\)
Convergence Type: Convergent
Sum: \(\pi^2/6 \approx 1.644934\)
Description: The sum can be bounded using the integral test to be within 1 and 2. The quest to find the exact sum of this series is known as the Basel problem, and was solved by Euler in 1735.

\[ \sum_{n=1}^\infty e^{-n} \]

First Few Terms: \(\frac{1}{e} + \frac{1}{e^2} + \frac{1}{e^3} + \frac{1}{e^4} + \cdots\)
Convergence Type: Convergent
Sum: \(\approx 0.581977\)
Description: Another series that can be tested with the integral test. Using the integral test with \(\int e^{-x} \,dx\) shows that this series converges.

\[ \sum_{n=1}^\infty \frac{1}{\sqrt{n}} \]

First Few Terms: \(1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} + \cdots\)
Convergence Type: Divergent
Sum: \(\infty\)
Description: A \(p\)-series with \(p = 1/2\). Diverges because \(p \le 1\).

\[ \sum_{n=1}^\infty ne^{-n^2} \]

First Few Terms: \(\frac{1}{e} + \frac{2}{e^4} + \frac{3}{e^9} + \frac{4}{e^{16}} + \cdots\)
Convergence Type: Convergent
Sum: \(\approx 0.404881\)
Description: A more complicated integral test example. Using the integral test with \(\int xe^{-x^2} \,dx\) shows that this series converges.

Floor 6

Comparison Tests

Some series look scary at first, but you soon realize that they’re just minor derivatives of familiar series.

\[ \sum_{n=1}^\infty \frac{1}{n^2 + 1} \]

First Few Terms: \(\frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} + \cdots\)
Convergence Type: Convergent
Sum: \(\approx 1.076674\)
Description: A direct comparison to the \(p\)-series \(\sum_{n=1}^\infty 1/n^2\) shows that this series converges.

\[ \sum_{n=1}^\infty \frac{2^n - 1}{3^n + 1} \]

First Few Terms: \(\frac{1}{4} + \frac{3}{10} + \frac{7}{28} + \frac{15}{82} + \cdots\)
Convergence Type: Convergent
Sum: \(\approx 1.371140\)
Description: A direct comparison to the geometric series \(\sum_{n=1}^\infty \left(\frac{2}{3}\right)^n\) shows that this series converges.

\[ \sum_{n=2}^\infty \frac{1}{n^2 - 1} \]

First Few Terms: \(\frac{1}{3} + \frac{1}{8} + \frac{1}{15} + \frac{1}{24} + \cdots\)
Convergence Type: Convergent
Sum: \(3/4 = 0.75\)
Description: A limit comparison to the series \(\sum_{n=1}^\infty 1/n^2\) shows that this series converges.
When the Direct Comparison Test is too weak, its stronger sibling, the Limit Comparison Test, can come to the rescue.

\[ \sum_{n=1}^\infty \frac{2^n + 1}{3^n - 1} \]

First Few Terms: \(\frac{3}{2} + \frac{5}{8} + \frac{9}{26} + \frac{17}{80} + \cdots\)
Convergence Type: Convergent
Sum: \(\approx 3.085607\)
Description: A limit comparison to the geometric series \(\sum_{n=1}^\infty \left(\frac{2}{3}\right)^n\) shows that this series converges.

\[ \sum_{n=1}^\infty \frac{1}{n^2 + \sin^2(n)} \]

First Few Terms: \(\frac{1}{1 + \sin^2(1)} + \frac{1}{4 + \sin^2(2)} + \frac{1}{9 + \sin^2(3)} + \frac{1}{16 + \sin^2(4)} + \cdots\)
Convergence Type: Convergent
Sum: \(\approx 1.183048\)
Description: A direct comparison to the \(p\)-series \(\sum_{n=1}^\infty 1/n^2\) shows that this series converges. Must use the fact that \(\sin^2(x) \ge 0\) for all \(x\).

Floor 7

Alternating Series

These series like to be fancy with their alternating signs. Despite this, you are undeterred from finding their properties.

\[ \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \]

First Few Terms: \(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots\)
Convergence Type: Conditionally Convergent
Sum: \(\ln(2) \approx 0.693147\)
Description: The alternating harmonic series - a classic example of an alternating series.
There seems to be something natural about the sum of this series...

\[ \sum_{n=0}^\infty (-1)^n\left[1 + \left(\frac{1}{2}\right)^n\right] \]

First Few Terms: \(2 - \frac{3}{2} + \frac{5}{4} - \frac{9}{8} + \cdots\)
Convergence Type: Divergent
Sum: Does Not Exist
Description: A series that fails the limit condition of the Alternating Series Test and also fails the Divergence Test.

\[ \sum_{n=2}^\infty \frac{(3/2)(-1)^n + 1/2}{\lfloor n/2 \rfloor} \]

First Few Terms: \(\frac{2}{1} - \frac{1}{1} + \frac{2}{2} - \frac{1}{2} + \frac{2}{3} - \frac{1}{3} + \cdots\)
Convergence Type: Divergent
Sum: \(\infty\)
Description: A series that fails the decreasing condition of the Alternating Series Test.

\[ \sum_{n=1}^\infty (-1)^n\frac{4}{2n+1} \]

First Few Terms: \(4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} \cdots\)
Convergence Type: Conditionally Convergent
Sum: \(\pi \approx 3.141593\)
Description: Another series that can be tested using the Alternating Series Test.
You don’t know why yet, but the number \(\pi\) appears in this sum. Your curiosity starts to grow...

\[ \sum_{n=1}^\infty \frac{\cos(n\pi)}{n^2} \]

First Few Terms: \(\frac{\cos(\pi)}{1} + \frac{\cos(2\pi)}{4} + \frac{\cos(3\pi)}{9} + \frac{\cos(4\pi)}{16} + \cdots\)
Convergence Type: Absolutely Convergent
Sum: \(\approx -0.822467\)
Description: A series that doesn’t look like an alternating series at first until you realize that \(\cos(n\pi) = (-1)^n\). Also an example of an absolutely convergent series.
Alternating series can take on multiple forms.

Floor 8

Ratio/Root Tests

The secret to finding these series’ convergence lies in an old friend, geometric series. Inspired by them, you create a pair of new convergence tests.

\[ \sum_{n=0}^\infty \frac{1}{n!} \]

First Few Terms: \(1 + 1 + \frac{1}{2} + \frac{1}{6} + \cdots\)
Convergence Type: Convergent
Sum: \(e \approx 2.718282\)
Description: A series involving factorials. The ratio test comes in useful here.
Yet another important constant mysteriously appears in this series’ sum. You become increasingly motivated to figure out why these constants keep appearing...

\[ \sum_{n=1}^\infty \left(\frac{2n^2 + 4n + 7}{n^2 - 2n - 17}\right)^n \]

First Few Terms: \(-\frac{13}{18} + \left(-\frac{23}{17}\right)^2 + \left(-\frac{37}{14}\right)^3 + \left(-\frac{55}{9}\right)^4 + \cdots\)
Convergence Type: Divergent
Sum: \(\infty\)
Description: The root test is useful for series like this one, where a complicated expression is raised to a power involving \(n\).
You feel a strange sense of familiarity with this series.

\[ \sum_{n=1}^\infty \frac{n}{2^n} \]

First Few Terms: \(\frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16} + \cdots\)
Convergence Type: Convergent
Sum: 2
Description: Another ratio test example, this time involving exponents. Interestingly, this sum can be calculated exactly by rewriting the series as a double summation.

\[ \sum_{n=1}^\infty \frac{n!}{n^n} \]

First Few Terms: \(\frac{1}{1} + \frac{2}{4} + \frac{6}{27} + \frac{24}{256} + \cdots\)
Convergence Type: Convergent
Sum: \(\approx 1.879854\)
Description: Another ratio test example that requires using L’Hôpital’s rule. The ratio obtained from the ratio test is \(1/e\).
A battle between factorials and exponents.

Floor 9

Power Series

The idea of infinite polynomials might sound terrifying. However, you might be able to use your knowledge of series to turn them into something much simpler.

\[ \sum_{n=0}^\infty x^n \]

First Few Terms: \(1 + x + x^2 + x^3 + \cdots\)
Convergence Type: Absolutely convergent if \(|x| \lt 1\)
Interval of Convergence: \((-1, 1)\)
Radius of Convergence: 1
Sum: \(\frac{1}{1-x}\) if \(|x| \lt 1\)
Description: The simplest power series possible - the geometric series sum formula comes in handy.

\[ \sum_{n=0}^\infty (2x^2)^n \]

First Few Terms: \(1 + 2x^2 + 4x^4 + 8x^6 + \cdots\)
Convergence Type: Absolutely convergent if \(|x| \lt \frac{\sqrt{2}}{2}\)
Interval of Convergence: \((-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\)
Radius of Convergence: \(\frac{\sqrt{2}}{2}\)
Sum: \(\frac{1}{1-2x^2}\) if \(|x| \lt \frac{\sqrt{2}}{2}\)
Description: A power series with a radius of convergence other than 1.

\[ \sum_{n=0}^\infty n!x^n \]

First Few Terms: \(1 + x + 2x^2 + 6x^3 + \cdots\)
Convergence Type: Absolutely convergent if \(x = 0\)
Interval of Convergence: \(x = 0\)
Radius of Convergence: 0
Sum: \(0\) if \(x = 0\)
Description: A power series with a radius of convergence of 0.
A single point of convergence.

\[ \sum_{n=1}^\infty \frac{x^n}{n^n} \]

First Few Terms: \(x + \frac{x^2}{2} + \frac{x^3}{27} + \frac{x^4}{256} +\cdots\)
Convergence Type: Absolutely convergent for all \(x\)
Interval of Convergence: \((-\infty, \infty)\)
Radius of Convergence: \(\infty\)
Description: A power series with an infinite radius of convergence.

\[ \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} \]

First Few Terms: \(1 - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\)
Convergence Type: Absolutely convergent if \(|x| \lt 1\)
Interval of Convergence: \((-1, 1)\)
Radius of Convergence: 1
Sum: \(\arctan(x)\) if \(|x| \lt 1\)
Description: Finding the sum of this series requires integrating the power series for \(\frac{1}{1+x^2}\).
That \(2n+1\) term does seem oddly familiar though...

\[ \sum_{n=1}^\infty \frac{-(-x)^n}{n} \]

First Few Terms: \(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} \cdots\)
Convergence Type: Absolutely convergent if \(-1 \lt x \le 1\)
Interval of Convergence: \((-1, 1]\)
Radius of Convergence: 1
Sum: \(\ln(1+x)\) if \(-1 \lt x \le 1\)
Description: Finding the sum of this series requires integrating the power series for \(\frac{1}{1+x}\).
You get flashbacks to the alternating harmonic series, and it all makes sense now.

Floor 10

Taylor Series

The deepest secrets of math lie here. Understanding these series grants you the godlike power to calculate almost anything.

\[ \sum_{n=0}^\infty \frac{x^n}{n!} \]

First Few Terms: \(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\)
Convergence Type: Absolutely convergent for all \(x\)
Interval of Convergence: \((-\infty, \infty)\)
Radius of Convergence: \(\infty\)
Sum: \(e^x\) for all \(x\)
Description: The Taylor series for \(e^x\).
It’s all starting to make sense now...

\[ \sum_{n=0}^\infty \frac{(-1)^n (x^{2n})}{(2n)!} \]

First Few Terms: \(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\)
Convergence Type: Absolutely convergent for all \(x\)
Interval of Convergence: \((-\infty, \infty)\)
Radius of Convergence: \(\infty\)
Sum: \(\cos(x)\) for all \(x\)
Description: The Taylor series for \(\cos(x)\).
You start to feel a wave of enlightenment...

\[ \sum_{n=0}^\infty \frac{(-1)^n (x^{2n+1})}{(2n+1)!} \]

First Few Terms: \(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\)
Convergence Type: Absolutely convergent for all \(x\)
Interval of Convergence: \((-\infty, \infty)\)
Radius of Convergence: \(\infty\)
Sum: \(\sin(x)\) for all \(x\)
Description: The Taylor series for \(\sin(x)\).
You feel a strong connection between this series and the series for \(e^x\) and \(\cos(x)\)... Is everything coming together?

\[ \sum_{n=0}^\infty \frac{(ix)^n}{n!} \]

First Few Terms: \(1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \cdots \)
Convergence Type: Absolutely convergent for all \(x\)
Interval of Convergence: \((-\infty, \infty)\)
Radius of Convergence: \(\infty\)
Sum: \(e^{ix} = \cos(x) + i\sin(x)\) for all \(x\)
Description: The Taylor series for \(e^{ix}\). Allows us to define complex exponentiation through Euler’s formula.
The moment has come. Out of curiosity, you plug in \(x = \pi\) into this series. As you make the shocking discovery that \(e^{i\pi} = -1\), you feel a sense of divine enlightenment. Everything has come together: Euler’s number \(e\), the imaginary unit \(i\), the circle constant \(\pi\)... You get flashbacks to all of your years of studying math, and you realize that much of it is contained in this one equation.

\[ \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)n!} \]

First Few Terms: \(x - \frac{x^3}{3} + \frac{x^5}{5(2!)} - \frac{x^7}{7(3!)} + \cdots \)
Convergence Type: Absolutely convergent for all \(x\)
Interval of Convergence: \((-\infty, \infty)\)
Radius of Convergence: \(\infty\)
Sum: \(\int_0^x e^{-t^2}\,dt\) for all \(x\)
Description: The Taylor series for the integral of \(e^{-x^2}\). Very useful in statistics for finding the area under a bell curve.
You recall seeing the function \(e^{-x^2}\) and was frustrated at not being able to find an antiderivative. With the power of infinite series, you have finally gotten your revenge.

\[ \sum_{n=1}^\infty \frac{(-1)^{n+1} (x-1)^n}{n} \]

First Few Terms: \((x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} + \cdots \)
Convergence Type: Absolutely convergent for all \(x\)
Interval of Convergence: \((0, 2]\)
Radius of Convergence: 1
Sum: \(\ln(x)\) for \(0 \lt x \le 2\)
Description: The Taylor series for \(\ln(x)\) centered at \(x = 1\).
As powerful as Taylor series are, they have their limitations - not all of them have an infinite radius of convergence.