Infinite Series and the Geometric Series
Define the sum of a series as the limit of its partial sums, work out the geometric and telescoping cases exactly, and use the term test to rule out convergence.
What a learner can do afterwards
- Write the partial sum of a telescoping series and take its limit
- State the condition under which a geometric series has a finite sum
- Show that terms not tending to zero force divergence, and that the converse fails
1 · Read
Let S_N mean the sum of the first N terms. A convergent series has a finite sum equal to the limit of S_N as N grows. For example, 1/(N + 2) tends to 0. Starting at n = 1, split 1/[(n + 1)(n + 2)] as 1/(n + 1) - 1/(n + 2). Neighbouring pieces cancel, leaving S_N = 1/2 - 1/(N + 2). Its limit is 1/2, the series sum.
For a geometric series, a is the first term and r the fixed real ratio. S_N = a(1 - r^N)/(1 - r) when r is not 1; when r = 1, S_N = Na. With nonzero a, convergence holds exactly when |r| < 1, with sum a/(1 - r). If a = 0, the sum is 0 for any real r. For 3 + 1 + 1/3 + ..., r = 1/3 and the sum is 3/(2/3) = 9/2. Its partial sums begin 3, 4, 13/3, 40/9.
The formula a/(1 - r) works for positive or negative ratios with |r| < 1. Identify a and r, calculate 1 - r, then divide a by that denominator. For 2 - 1 + 1/2 - 1/4 + ..., a = 2 and r = -1/2. The denominator is 1 - (-1/2) = 3/2, so the sum is 2/(3/2) = 4/3. Alternating signs do not prevent convergence.
The term test proves divergence whenever the terms do not tend to zero. Rafi sees a series whose nth term tends to 7 and correctly declares divergence. The test also applies if the terms have no limit, including terms that repeatedly return to zero but do not tend to zero. Terms tending to zero are necessary for convergence; the term test alone cannot prove convergence.
The harmonic series is 1 + 1/2 + 1/3 + ... Its terms tend to zero, but its partial sums grow without bound. Group consecutive terms: 1/3 + 1/4 > 1/2, and 1/5 + 1/6 + 1/7 + 1/8 > 1/2. Later blocks with twice as many terms also add more than 1/2. This gives a counterexample: terms tending to zero do not guarantee convergence.
For a grouping proof, take harmonic terms from n = 2^k + 1 to n = 2^(k + 1), for k = 0, 1, 2, ... Each block has 2^k terms, each at least 1/2^(k + 1). Its sum is therefore at least 1/2. After m blocks, the partial sum is at least 1 + m/2. Since this bound grows without limit, the partial sums cannot approach a finite value.
Try cancellation when neighbouring pieces might disappear. For a geometric series, check its first term and ratio. Use the term test to prove divergence when terms do not tend to zero. If they tend to zero, another argument is needed.
Partial sums define a convergent series' sum. Check a geometric series' first term and ratio. Terms tending to zero alone do not prove convergence.
Readings and practice questions are original. Video links are from Khan Academy.
2 · Watch
Take it off screen
Where it sits
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.