For each of the following, determine if the series converges. Be sure that you c

Question

For each of the following, determine if the series converges.Be sure that you can rigorously demonstrate your answer!

For each of the following, determine if the series converges.Be sure that you can rigorously demonstrate your answer! 1.displaystyle sumlimits_{n=5}^{infty} {nover n^{4} + 10 n^{2} + 25} 2.displaystyle sumlimits_{n=1}^{infty} frac{1}{ln(5^{n})} 3.displaystyle sumlimits_{n=0}^{infty} {5over n^2 + 5} 4.displaystyle sumlimits_{n=5}^{infty} {n + 5over n^2 + 10 n + 5} 5.displaystyle sumlimits_{n=1}^{infty} {5over (5 n - 1)^{5}} 6.displaystyle sumlimits_{n=1}^{infty} {5 nover (5 n + 1)} 7.displaystyle sumlimits_{n=1}^{infty} {n + 5^nover n 5^n}

Explanation / Answer

1.) Converges. You should use the limit comparison test with 1/n3

2.) Diverges. Note ln(5^n) = nln(5). use limit comparison test with 1/n

3.) Converges. limit comparison test with 1/n2

4.) Diverges. limit comparison test with 1/n

5.) Converges.

6.) Divergent. Use the test for divergence.

7.) Diverges

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