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State if the following series converges or diverges

sum[k=1,inf] cos(k)/(k^2+1)

I applied the convergence test

lim k->inf cos(k)/(k^2+1) =H lim k->inf -sin(k)/(2k) =H lim k->inf -cos(k)/2 = some undefined value bounded by -1 and 1 =/= 0 so the series diverges by the divergence test. I guess my answer is wrong.

This question came up on a assignment in class. We went over the answer after handing in the the assignment. The lecturer stated that the the series did not converge and compared it to sum[k=1,inf] 1/k^2 and stated that

0 < sum[k=1,inf] cos(k)/(k^2+1) < sum[k=1,inf] 1/k^2

and because sum[k=1,inf] 1/k^2 is convergent by the p-series test, 2>1, sum[k=1,inf] cos(k)/(k^2+1) must also converge by the comparison test sense sum[k=1,inf] cos(k)/(k^2+1) < sum[k=1,inf] 1/k^2.

I believed this to be true at the time this was stated. I however have a problem accepting this because

0 < sum[k=1,inf] cos(k)/(k^2+1)

is not true statement because cos(k) alternates between positive values and negative values.

I also have a problem with this statement

sum[k=1,inf] cos(k)/(k^2+1) < sum[k=1,inf] 1/k^2

cos(k) is bounded by positive and negative one and is undefined as k goes to infinity but stuck in between these two values. As k goes to infinity

k^2+1 = k^2

so the denominators would be equal to each other

and at some very large values cos(k) = 1

so wouldn't this statement be more correct

sum[k=1,inf] cos(k)/(k^2+1) <= sum[k=1,inf] 1/k^2

and I'm not sure the comparison test works here when you have >= or <= comparisons. I thought they had to be strictly < or >

Just from thinking about it... I would like to believe that we cannot say anything at all about the series with comparison tests with >= or <=... but I'm not so sure as to exactly why at the moment...

I checked with wolfram alpha

http://www.wolframalpha.com/input/?i=sum[n=1,+inf]+of+cos(k)/(k^2+1)"

and it says the series does not converge by the limit test as I showed. I however know computers can be wrong sometimes and have seen wolfram alpha return wrong results before, but I'm not so sure that the series converges and don't see any test that I know of that would be practical to use besides the comparison test or the limit comparison test in which it's hard to state < or > or <= or >= because cos(k) alternates between positive one and negative one and is undefined as k goes to infinity but and is stuck some were in between positive and negative one including those values.

Thanks for any help.

sum[k=1,inf] cos(k)/(k^2+1)

I applied the convergence test

lim k->inf cos(k)/(k^2+1) =H lim k->inf -sin(k)/(2k) =H lim k->inf -cos(k)/2 = some undefined value bounded by -1 and 1 =/= 0 so the series diverges by the divergence test. I guess my answer is wrong.

This question came up on a assignment in class. We went over the answer after handing in the the assignment. The lecturer stated that the the series did not converge and compared it to sum[k=1,inf] 1/k^2 and stated that

0 < sum[k=1,inf] cos(k)/(k^2+1) < sum[k=1,inf] 1/k^2

and because sum[k=1,inf] 1/k^2 is convergent by the p-series test, 2>1, sum[k=1,inf] cos(k)/(k^2+1) must also converge by the comparison test sense sum[k=1,inf] cos(k)/(k^2+1) < sum[k=1,inf] 1/k^2.

I believed this to be true at the time this was stated. I however have a problem accepting this because

0 < sum[k=1,inf] cos(k)/(k^2+1)

is not true statement because cos(k) alternates between positive values and negative values.

I also have a problem with this statement

sum[k=1,inf] cos(k)/(k^2+1) < sum[k=1,inf] 1/k^2

cos(k) is bounded by positive and negative one and is undefined as k goes to infinity but stuck in between these two values. As k goes to infinity

k^2+1 = k^2

so the denominators would be equal to each other

and at some very large values cos(k) = 1

so wouldn't this statement be more correct

sum[k=1,inf] cos(k)/(k^2+1) <= sum[k=1,inf] 1/k^2

and I'm not sure the comparison test works here when you have >= or <= comparisons. I thought they had to be strictly < or >

Just from thinking about it... I would like to believe that we cannot say anything at all about the series with comparison tests with >= or <=... but I'm not so sure as to exactly why at the moment...

I checked with wolfram alpha

http://www.wolframalpha.com/input/?i=sum[n=1,+inf]+of+cos(k)/(k^2+1)"

and it says the series does not converge by the limit test as I showed. I however know computers can be wrong sometimes and have seen wolfram alpha return wrong results before, but I'm not so sure that the series converges and don't see any test that I know of that would be practical to use besides the comparison test or the limit comparison test in which it's hard to state < or > or <= or >= because cos(k) alternates between positive one and negative one and is undefined as k goes to infinity but and is stuck some were in between positive and negative one including those values.

Thanks for any help.

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