Let ak and bk be series with positive terms. Prove:(a) If limk+ (ak/bk) = 0 and bk

Chapter 9, Problem 56

(choose chapter or problem)

Let $\sum a_{k}$ and $\sum b_{k}$ be series with positive terms. Prove:

(a) If $\lim _{k \rightarrow+\infty}\left(a_{k} / b_{k}\right)=0$ and $\sum b_{k}$ converges, then $\sum a_{k}$ converges.

(b) If $l i m_{k \rightarrow+\infty}\left(a_{k} / b_{k}\right)=+\infty$ and $\sum b_{k}$ diverges, then $\sum a_{k}$ diverges.

Equation Transcription:

$\sum\limits_{\ }^{\ }{{a}_{k}}^{\ }$

$\sum\limits_{\ }^{\ }{{b}_{k}}^{\ }$

${lim}_{k\rightarrow{}+\infty{}}({a}_{k}/{b}_{k})=0$

$\sum\limits_{\ }^{\ }{{b}_{k}}^{\ }$

$\sum\limits_{\ }^{\ }{{a}_{k}}^{\ }$

${lim}_{k\rightarrow{}+\infty{}}({a}_{k}/{b}_{k})=+\infty{}$

$\sum\limits_{\ }^{\ }{{b}_{k}}^{\ }$

$\sum\limits_{\ }^{\ }{{a}_{k}}^{\ }$

Text Transcription:

Sum a_k

Sum b_k

Lim_k rightarrow +infty (a_k/b_k)=0

Sum b_k

Sum a_k

lim_k rightarrow +infty (a_k/b_k)=+infty

Sum b_k

Sum a_k

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