This problem deals with the integral test however, do NOT write down or do any i

Question

This problem deals with the integral test however, do NOT write down or do any integrations NOR state if these series converge or diverge. Instead, for each series explicitly STATE what function (i.e. f(x) = . . . ) you are considering then answer YES if the integral test may be used or NO if it may not be used. In either case JUSTIFY your response; for the YES case confirming and/or showing why all three hypothesis (of the integral test) are met and for the NO case stating/showing which hypothesis your stated f(x) function fails to meet.

(a) series from k=1 to inf of e^k/k^2

(b) series from k=1 to inf of (ln(1/k))/(1/k^2)

(c) series from k=1 to inf of ((pi/2)-tan^-1(k))/(k^2+1)

Explanation / Answer

Ans)

a) series from k=1 to inf of e^k/k^2

finding ratio test, lim n tends to infinty |an+1/an| = lim n tends to infinty(|(e^(k+1)/(k+1)^2)*(k^2/e^k)|)

                                                                    = lim n tends to infinty(|(e(k^2)/(k+1)^2)|) = e i.e., diverges

We dont use integral test.

b)series from k=1 to inf of (ln(1/k))/(1/k^2)

if lim n tends to infinity an not equal to 0 then sigma(an) diverges

lim n tends to infinity( (ln(1/k))/(1/k^2)) = -infinity

here also we dont use integral test and this series diverges.

c) series from k=1 to inf of ((pi/2)-tan^-1(k))/(k^2+1)

from comparision test

series from k=1 to inf of ((pi/2)-tan^-1(k))/(k^2+1) = series from k=1 to inf of ((pi/2))/(k^2+1) - series from k=1 to inf of (-tan^-1(k))/(k^2+1)

    by comparision test it converges.

Here also we dont use integral test

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