Definitions and Theorems-Calculus III
Infinite Series
Def. A sequence {an} converges if exists (and is finite). If the sequence has no limit (goes to infinity, oscillates, etc.) then it diverges.
Def. The partial sums of the seriesare:
S1 = a1
S2 = a1 + a2
S3 = a1 + a2 + a3
etc.
Def. The series converges if the sequence of partial sums has a limit. If this sequence has no limit (goes to infinity, oscillates, etc.) then the series diverges.
Note A sequence can converge even though the associated series diverges. For example
the sequence {1,1,1,1, …} converges to 1 but the series 1 + 1 + 1 + 1 + … diverges (approaches infinity).
Geometric series – The series diverges if │r │ 1,
converges if – 1 < r < 1. In the latter case the sum is .
When summing geometric series, you also need:
(assuming the component series are all convergent).
Harmonic series – The series diverges.
nth Term Test – If ≠ 0 (or the limitdoesn’t exist), then the series diverges. (This can also be written as: if the series converges, then the limit of the terms must be zero.)
Integral Test – Assume the series satisfies:
1) All an > 0
2) The sequence {an} is decreasing
3) = 0
Then the series converges if and only if converges, where f(n) = an.
p-series The series converges if p > 1, diverges if 0 < p < 1.
Limit Comparison Test — Assume andare positive series.
If where L is a finite, nonzero constant, then the two series both converge or both diverge (one series converges if and only if the other series converges).
Alternating Series Test —Assume {bn} satisfies
i) bn > 0 for all n
ii) {bn} is decreasing
iii) = 0
Then the alternating series b1 ‒ b2 + b3 ‒ … is convergent.
Ratio Test
Let
i) If 0 ρ < 1, then the series is absolutely convergent (AC).
ii) If ρ > 1 or ρ = ∞, then is divergent (D).
iii) If ρ = 1 then Ratio Test fails, you need to use another test. (Usually you need to use Limit Comparison Test.)
Math 260, theorems on series
Page 2 of 2, Spring 2010