Question

For each series, decide whether the series converges (absolutely or conditionally) or diverges. If able, evaluate the sum 9fisn what it coverges to) Show all work.

A)

$\sum_{k=1}^{inifinity} \sqrt{k^4+2})/k^2$

b)

$\sum_{k=1}^{infinity} 2e^{-3k}$

c)

$\sum_{k=3}^{infinity} e^{^{k}}(k+1)/(k^{2}k!)$

d)$\sum_{k=3}^{infinity} (-1)^{^{k+1}}lnk/(\sqrt{k-1})$

Answer 1

"As we know an infinite positive term series Rightarrow Sigma^infinity_R = 1 U_R converges iff lim_R rightarrow infinity U_R = 0 Sigma^infinity_k = 1 squareroot k^4 + 2/k^2 U_k = squareroot k^4 + 2/k^2 = squareroot 1 + 2/k^2 Here lim_k rightarrow infnity U_x = 1 notequalto 0 Rightarrow u_k is not convergent series Sigma^infinity_k = 1 2 e^-3k rightarrow (This is a positive term series Here u_k = 2 e^-3k lim_k rightarrow infinity u - K = 0 (Sigma U_x satisfies necessary condition) Now Apply D'alembert Ratio test Sigma_k rightarrow infinity u_x/u_k + 1 = lim_k rightarrow infinity 2 e^-3k/2 e^-3 (k + 1) e^3 > 1 Rightarrow Sigma u_k converges also Sigma