i need help with two questions for my calculus about sequences and series

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Apr 26th, 2015

f) The series ∑j=1(–1)j sin j / j3 converges by the comparison test, because |(–1)j sin j / j3| ≤ 1/ j3 and the series ∑j=11/ jp converges for any p > 1. Here p = 3.

g) The series ∑n=21/n(ln n)2 converges since the integral ∫21/x(ln x)2 dx = – 1/ln x ]2= 1/ln 2 converges (note that limx→+∞1/ln x = 0).


Apr 27th, 2015

I noticed that you have also asked about your third question (e) and got a wrong answer. Here is a solution.

Apply the ratio test to the series ∑n=0 [n6n+1 / n!5].

limn→ an+1 /an = limn→∞ [(n+1)6n+2 / (n+1)!5n+1] ÷ [n6n+1 / n!5n] =

limn→∞ [(n+1)·n!·5n·6n+2 / n·(n+1)!·5n+1·6n+1] = limn→∞ 6/(5n) = 0 < 1 and the series converges.



Apr 29th, 2015

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