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.
k=110k
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Expert Answer

Theodore Schwartz
Answered 2021-02-06 Author has 99 answers

Given, the series is
k=110k
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 ak is an infinite series, then Sk=a1+a2+a3+...+ak is called the n partial sum
2.If ak is an series of positive terms s.t. limkakak+1=l, then the series ak is divergent if l<1
Now, to find the first four terms of a sequence of partial sums
ak=10k
S1=a1=101=10
S2=a1+a2=101+102=10+100=110
S3=a1+a2+a3=101+102+103=10+100+1000=1110
S4=a1+a2+a3+a4=101+102+103+104=10+100+1000+10000=11110
Since
S1=10
S2=110
S3=1110
S4=11110

Sk=k=010k
Now,
ak=10k, ak+1=10k+1
limk10k10k+1=lim110=110<1
Hence the series is divergent.

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Jeffrey Jordon
Answered 2021-12-25 Author has 2262 answers

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