convergence tests need help!

[johnq2k7]

Determine whether each of the series below is divergent, absolutely convergent (hence convergent) or conditionally convergent. Indicate the test, result or results you use to support you conclusions.

a.) sigma notation from (k=1 to infinity) of sqrt(k)/ ((sqrt(k)+3))

b.) sigma notation from (k=1 to infinity) of ((-1)^k)/((sqrt(k(k+1)))

c.) sigma notation from (k=1 to infinity) of (3*k^2)-1/ (k^4)

d.) sigma notation from (k=1 to infinity) of tan^-1(k)/ (k^2)

e.) sigma notation from (n=1 to infinity) of ((-1)^(n+1))*{((3^(2n-1)) / ((n^2)+1)}

f.) sigma notation from (k=1 to infinity) of ((5^k)+k)/(k!+3)

my work:


for a; if i divide by k i get the limit equals to 1 since 3/inf it converges to zero ....what theorem is this a comparison or alternating series

for b: it is a harmonic series because of (-1)^n therefore it alternates between + and - values so it diverges

for c. if i divide by k^4 it converges to zero

for d. it becomes tan^-1(0)/k.....so it converges to zero

for e. i need a lot of help here

for f; i need a lot of help here

I need a lot of help with these problems.. I don't know which theorem, test to use to find whether it converges, diverges or conditionally converges

i believe i need to use the limit comparison test, alternating sequences test, harmonic series test, partial ratio test, the roots test... but i need a lot of help applying it here... please help me with these problems...

please help!

thanks for all your assistance

 

[fasteddie65]

I'll pick one: ? from (k=1 to infinity) of tan^-1(k)/ (k^2)

tan^-1 k/k^2 > tan^-1 k/(k^2 + 1)

Integral Test: int [tan^-1 x/(x^2 + 1) dx = int [u du] = u^2/2 = (tan^-1 x)^2/2, which diverges as x --> ?, so the series also diverges.

 

[johnq2k7]

I'll pick one: ? from (k=1 to infinity) of tan^-1(k)/ (k^2)

tan^-1 k/k^2 > tan^-1 k/(k^2 + 1)

Integral Test: int [tan^-1 x/(x^2 + 1) dx = int [u du] = u^2/2 = (tan^-1 x)^2/2, which diverges as x --> ?, so the series also diverges.

thanks for your help... how about a through c is my approach correct and for e