Math Challenges

Submissions for Problem #54

Problem #54

Which of the following are convergent series? Justify your answer.

\[ \displaylines{\text{a) }\sum_{k=1}^{\infty}\cos\left(\left(2k+1\right)\frac{\pi}{2}\right)\\ \text{b) }\sum_{k=1}^{\infty}\frac{4^{k-1}}{5^{k}}\\ \text{c) }\sum_{k=1}^{\infty}\frac{3^{k}}{2^{k}}} \]

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zapwai

Solution: \( \displaylines{\text{Since }\cos\left(\frac{\left(2k+1\right)\pi}{2}\right)=0,\text{ we have the sum of zero for part (a).}\\ \text{For (b), }\frac15\sum_{k=0}^{\infty}\left(\frac45\right)^{k}=\frac15\left(\frac{1}{1-\left(\frac45\right)}\right)=1,\text{ again convergent.}\\ \text{For (c), this is divergent, as it is geometric with a base larger than one.}} \)