Series Tests Geometric Series: Σ ar.-a+ ar + ar' +..tar" + (witha # 0) converges if I < 1 and vrges f 21 Divergence Test: If lima,#0, Σα.dverges. Integral Test: Suppose fis continuous, positive, decreasing function (at least eventually) with 4° f(n). Then, if If(x)ax converges, Σα. converges; if I f(x)ax dverges, Σα. diverges. p-series: Σ +-+-+-+.. converges if p > 1 and diverges if 0 < p 1 The comparison tests must have positive terms (1) Direct Comparison b. and Σ>, converges, then Σ a, converges. If a, If b, s a, and Σ4, diverges, then Σ a, diverges. (2) The Limit Comparison Test: If lim - 1 andan fin, then both Ya and b, behave the same. That is, both series converge or both diverge Alternating Series Test for either form Σ(-1)"'q or Σ(-1)"a, (1) Ignoring the + signs, check that the terms are decreasing. That is, make certain that (2) Check that the terms are heading to zero. That is, make sure that lima-o If both conditions are satisfied, then the alternating series converges. The Ratio Test for Absolute Convergence lflimPatil-L and L < 1 , then the series Σ a, converges absolutely. If L > 1 , then the series diverges. If L = 1 , then the test provides no useful information. The Root Test for Absolute Convergence !flim Vla,-L and L < 1 . then the series Σ a, converges absolutely. If L > 1 , then the series diverges. If L = 1 , then the test provides no useful information.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 50E

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Question

Hi,

Here's my question:

Determine if the series is absolutely convergent, conditionally convergent, or divergent.

(summation of n=1 and n goes to infinity of) sin(pi(n)/6)/(1+n^(3/2))

This series converges by the Direct Comparison Test (see photo of a list of series tests below), but I need to use another test or two to find if the series is conditionally or absolutely convergent. However, I don't know which one in the list to use. All I know is that I can't use the root test, and it doesn't seem like the integral test will work.

Thanks!

Series Tests
Geometric Series: Σ ar.-a+ ar + ar' +..tar" + (witha # 0) converges if I < 1 and
vrges f 21
Divergence Test: If lima,#0, Σα.dverges.
Integral Test: Suppose fis continuous, positive, decreasing function (at least eventually) with
4° f(n). Then, if If(x)ax converges, Σα. converges; if I f(x)ax dverges, Σα. diverges.
p-series: Σ +-+-+-+.. converges if p > 1 and diverges if 0 < p 1
The comparison tests must have positive terms
(1) Direct Comparison
b. and Σ>, converges, then Σ a, converges.
If a,
If b, s a, and Σ4, diverges, then Σ a, diverges.
(2) The Limit Comparison Test: If lim
- 1 andan fin, then both
Ya and
b, behave the same. That is, both series converge or both diverge
Alternating Series Test for either form Σ(-1)"'q or Σ(-1)"a,
(1) Ignoring the + signs, check that the terms are decreasing. That is, make certain that
(2) Check that the terms are heading to zero. That is, make sure that lima-o
If both conditions are satisfied, then the alternating series converges.
The Ratio Test for Absolute Convergence
lflimPatil-L and L < 1 , then the series Σ a, converges absolutely. If L > 1 , then the
series diverges. If L = 1 , then the test provides no useful information.
The Root Test for Absolute Convergence
!flim Vla,-L and L < 1 . then the series Σ a, converges absolutely. If L > 1 , then the
series diverges. If L = 1 , then the test provides no useful information.

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