 # For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges. sum_{k=1}^infty10^k Alyce Wilkinson 2021-02-05 Answered
For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.
$\sum _{k=1}^{\mathrm{\infty }}{10}^{k}$
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Given, the series is
$\sum _{k=1}^{\mathrm{\infty }}{10}^{k}$
We have to find the first four terms of the sequence of partial sums, make a conjecture and state that the series is divergent.
1. If $\sum {a}_{k}$ is an infinite series, then ${S}_{k}={a}_{1}+{a}_{2}+{a}_{3}+...+{a}_{k}$ is called the n partial sum
2.If $\sum {a}_{k}$ is an series of positive terms s.t. $\underset{k\to \mathrm{\infty }}{lim}\frac{{a}_{k}}{{a}_{k+1}}=l$, then the series $\sum {a}_{k}$ is divergent if $l<1$
Now, to find the first four terms of a sequence of partial sums
${a}_{k}={10}^{k}$
${S}_{1}={a}_{1}={10}^{1}=10$
${S}_{2}={a}_{1}+{a}_{2}={10}^{1}+{10}^{2}=10+100=110$
${S}_{3}={a}_{1}+{a}_{2}+{a}_{3}={10}^{1}+{10}^{2}+{10}^{3}=10+100+1000=1110$
${S}_{4}={a}_{1}+{a}_{2}+{a}_{3}+{a}_{4}={10}^{1}+{10}^{2}+{10}^{3}+{10}^{4}=10+100+1000+10000=11110$
Since
${S}_{1}=10$
${S}_{2}=110$
${S}_{3}=1110$
${S}_{4}=11110$
$⋮$
${S}_{k}=\sum _{k=0}^{\mathrm{\infty }}{10}^{k}\to \mathrm{\infty }$
Now,

$\underset{k\to \mathrm{\infty }}{lim}\frac{{10}^{k}}{{10}^{k+1}}=lim\frac{1}{10}=\frac{1}{10}<1$
Hence the series is divergent.

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