 # Determine whether three series converges or diverges. if it converges, find its sum sum_(n=0)^(oo)((-1)^n)/4^n Braxton Pugh 2021-02-21 Answered
Determine whether three series converges or diverges. if it converges, find its sum
$\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-1\right)}^{n}}{{4}^{n}}$
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1)Here the series is given by $\sum _{n=1}^{\mathrm{\infty }}{\left(-1\right)}^{n}{a}_{n}=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-1\right)}^{n}}{{4}^{n}}$.This is a alternating series with ${a}_{n}=\frac{1}{{4}^{n}}.C\le arly{a}_{n}$ is a decreasing sequence and
$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{4}^{n}}=0$
Therefore by Leibnit'z rule the given series $\sum _{n=1}^{\mathrm{\infty }}\frac{{\left(-1\right)}^{n}}{{4}^{n}}$ is convergent.
2)Consider the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.The n-th partial sum of the series is given by
${S}_{n}=1+\frac{1}{2}+\frac{1}{3}+\dots +\frac{1}{n}$.
Now
${S}_{{2}^{n}}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\dots +\frac{1}{{2}^{n}}$
$=1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)+\dots +\frac{1}{{2}^{n}}$
$⇒1+\frac{1}{2}+\left(\frac{1}{4}+\frac{1}{4}\right)+\left(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\right)+\dots +\left(\frac{1}{n}+\dots +{2}^{n-1}\frac{1}{{2}^{n}}\right)$
$=1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\dots +\frac{1}{2}$
$=1+\frac{n}{2}$
but $\left\{\frac{n}{2}\right\}$ is a divergent sequence. This shows that ${S}_{{2}^{n}}$
is also divergent and hence the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$ is divergent.