← Calculus II

Unit 16: Infinite Series

Adding infinitely many numbers without getting nonsense — which sums settle down, and which run away forever.

7 topics · 29 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Sigma Notation for a Series

  1. How it worksHow does a summation symbol with an index, a starting value, and a formula unpack into the actual terms?
  2. Worked exampleFind the value of the sum from n = 1 to 5 of n2.
  3. Worked exampleFind the value of the sum from n = 3 to 6 of (2n - 1).
  4. Common mistakeWhy doesn't changing the starting index of a sum from n = 0 to n = 1 leave the sum's value unchanged?
Your turn1 of 2
Find the value of the sum from n = 1 to 6 of n2.

Series and Partial Sums

  1. How it worksHow does the sequence of partial sums, S1, S2, S3, and on, decide whether an infinite series converges at all?
  2. Worked exampleFind the first four partial sums of the series 1 + 1/2 + 1/4 + 1/8 and describe where they are heading.
  3. Worked exampleFind the first four partial sums of the series 1 + 2 + 3 + 4 and describe what they are doing instead.
  4. Common mistakeWhy doesn't a series converge just because its terms are getting smaller and smaller?
  5. Going furtherWhat happens to a series' partial sums when the terms alternate in sign forever without shrinking to zero?
Your turn1 of 2
Find the fourth partial sum of the series 4 + 4/2 + 4/4 + 4/8.

The Geometric Series

  1. How it worksHow does multiplying a geometric sum by its ratio r and subtracting the original produce the formula a/(1 - r)?
  2. Worked exampleFind the sum of the geometric series 4 + 2 + 1 + 1/2 continuing forever.
  3. Worked exampleFind the total distance traveled by a ball dropped from 10 meters that rebounds to 70 percent each bounce.
  4. Common mistakeWhy doesn't a geometric series with a ratio equal to exactly 1 or -1 add up to a finite total?
  5. Going furtherWhat happens to the geometric series sum formula as the ratio r approaches 1 from below?
Your turn1 of 2
Find the partial sum Sn = a + a·r + a·r2 + ... + a·rn-1 of a geometric series with first term 6 and common ratio 0.46, as a function of n.

Telescoping Series

  1. How it worksHow does splitting 1/(n(n+2)) into a difference of two simpler fractions make the whole sum telescope?
  2. Worked exampleFind the sum from n = 1 to infinity of 1/(n(n + 2)).
  3. Common mistakeWhy doesn't every series built from a difference of fractions actually telescope down to something finite?
Your turn1 of 2
Find the sum from n = 1 to infinity of 4/(n(n + 2)).

The nth-Term Test for Divergence

  1. How it worksHow does the limit of the nth term give a one-way test that can only ever prove divergence, never convergence?
  2. Worked exampleShow me why the series with terms 3n/(2n + 1) has to diverge.
  3. Common mistakeWhy doesn't a series' terms shrinking to zero prove that the series itself converges?
  4. Going furtherWhat happens when the nth-term test comes back inconclusive because the limit of the terms is exactly zero?
Your turn1 of 2
Why does the series with terms 3n/(2n + 1) have to diverge?

The Harmonic Series

  1. How it worksHow does grouping the harmonic series into blocks that each add to at least one-half force the sum to diverge?
  2. Worked exampleFind roughly how many terms of the harmonic series it takes for the partial sum to first exceed 4.
  3. Common mistakeWhy isn't the harmonic series convergent even though its terms head to zero exactly as fast as a convergent p-series?
  4. Ties togetherHow does the harmonic series' divergence connect to the improper integral of 1/x from 1 to infinity?
Your turn1 of 2
Find roughly how many terms of the harmonic series it takes for the partial sum to first exceed 5, using the approximation Hn is about ln(n) + 0.5772.

p-Series

  1. How it worksHow does comparing a p-series to the improper integral of 1/xp draw the dividing line at exactly p = 1?
  2. Worked exampleFind whether the p-series sum of 1/n3/2 converges or diverges.
  3. Common mistakeWhy doesn't a p-series with p = 1 converge when a p-series with any larger p does?
  4. Going furtherWhat happens to a p-series when the exponent p is negative instead of positive?
Your turn1 of 2
Does the p-series sum of 1/n3/2 converge or diverge?