Convergence Test: sqrt(k)/ ((sqrt(k)+3)), ((-1)^k)/((sqrt(k(

[djdavis2k]

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 (n=1 to infinity) of ((-1)^(n+1))*{((3^(2n-1)) / ((n^2)+1)}

e.) 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 ....however the p-series is less than 1 so it diverges.. so it's conditionally convergent is this correct?

for b: it is a harmonic series because of (-1)^n therefore it alternates between + and - values so it diverges and the p-series of bn is less than 1 so it also diverges so it's divergent is this correct

for c. if i divide by k^4 it converges to zero and the absolute value of |an| also converges so it is absolutely convergent.. is this correct?

for d. if i divide the expressions by n^2 i'm left with 3^0= 1 and since lim of an does not approach 0..it diverges.. is this correct?

for e. i'm not sure what to do.. do i apply the ratio test.. and what do i with the value i obtain from it.. how do i know whether it is conditional or absolute convergence if it does converge?


i believe using the limit comparison test, alternating sequences test, harmonic series test, partial ratio test, the roots test and p-series helps but i want to verify if these answers from my work is correct

PLEASE HELP!
thanks for all your assistance

 

[daon]

Re: Convergence Test! A LOT OF WORK SHOWN...PLEASE HELP ME!!!!

For the ratio test, if the limit of \(\displaystyle | \frac{a_{n+1}}{a_n}| < 1\) then it converges absolutly. Greater than 1, it diverges.

 

[galactus]

e.)\(\displaystyle \sum_{k=1}^{\infty}\frac{5^{k}+k}{k!+3}\)

for e. i'm not sure what to do.. do i apply the ratio test.. and what do i with the value i obtain from it.. how do i know whether it is conditional or absolute convergence if it does converge?

The criterion for the ratio test is in any calc book.

Using the ratio test we have \(\displaystyle {\rho}=\lim_{k\to \infty}\frac{(k!+3)(5^{k+1}+k+1)}{(k!(k+1)!+3)(5^{k}+k)}=0\)

Since \(\displaystyle {\rho}<1\), it converges.

If \(\displaystyle {\rho}>1\), it diverges.

If \(\displaystyle {\rho}=1\), inconclusive and use another test.

 

[djdavis2k]

e.)\(\displaystyle \sum_{k=1}^{\infty}\frac{5^{k}+k}{k!+3}\)

[quote:2q1j3fkm]for e. i'm not sure what to do.. do i apply the ratio test.. and what do i with the value i obtain from it.. how do i know whether it is conditional or absolute convergence if it does converge?

The criterion for the ratio test is in any calc book.

Using the ratio test we have \(\displaystyle {\rho}=\lim_{k\to \infty}\frac{(k!+3)(5^{k+1}+k+1)}{(k!(k+1)!+3)(5^{k}+k)}=0\)

Since \(\displaystyle {\rho}<1\), it converges.

If \(\displaystyle {\rho}>1\), it diverges.

If \(\displaystyle {\rho}=1\), inconclusive and use another test.[/quote:2q1j3fkm]

how about b.?


for b: it is a harmonic series because of (-1)^n therefore it alternates between + and - values so it diverges and the p-series of bn is less than 1 so it also diverges so it's divergent is this correct?