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1)

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I applied Raabe's test on both and guessed the answer as (B), but not convinced enough. Is there a better approach?


2) Is there any closed form of following the series. It is however, known that the sum is irrational. (sorry for poor formatting)

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    It is not good that you only ask your question as images. As soon as your image hoster stops hosting your images, the question does not make any sense anymore.2011-05-07
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    For 2: [one of these four](http://dlmf.nist.gov/20.2.i) will be useful in determining the closed form.2011-05-07
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    Why did you decide to apply Raabe's test? Are the conditions of this test fulfilled? Do you know any simple properties that convergent series have?2011-05-07
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    I think for the first problem, both of them diverge, termwise the first series is same order as $\displaystyle \sum^{\infty}_{n=1} \frac{\pi}{n}$ when $n$ is large2011-05-07
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    @user9325: Could not agree more. Tried to raise concern about this (mal)practice, which should be banned for the reason you say and for others as well. Failed. See http://meta.math.stackexchange.com/questions/18052011-05-07

1 Answers 1

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Series $(ii)$ diverges, since the absolute value of its $n$th term converges to 1 rather than 0.

Series $(i)$ diverges as well. For $x$ small, $\sin(x)=x+o(x^2)$ (recall the Taylor series for $\sin(x)$ about 0), so the given series behaves like the harmonic series plus a convergent series.

There is no known closed form (in terms of elementary expressions) for the series. However, it can be expressed in terms of Jacobi's theta function as $\displaystyle\frac{\vartheta_3(0,1/2)-1}2$.

(A good idea when looking for closed form expressions for numerical series is to first try is the wonderful page for Plouffe's inverter.)

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    "There is no known closed form for the series. However, it can be expressed in terms of Jacobi's theta function..." - I got confused a bit there... probably you intended to say "there's no elementary expression..."2011-05-07
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    :-) J.M.: I think you commented before I edited, precisely because of the oddity.2011-05-07
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    Thank you all. Raabe's test is only applicable for positive term series. I was seriously mistaken!2011-05-07