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

Question
Series
asked 2021-02-05
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.
\(\sum_{k=1}^\infty10^k\)

Answers (1)

2021-02-06
Given, the series is
\(\sum_{k=1}^\infty10^k\)
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. \(\lim_{k\rightarrow\infty}\frac{a_k}{a_{k+1}}=l\), then the series \(\sum a_k\) is divergent if \(l<1\)</span>
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
\(S_1=10\)
\(S_2=110\)
\(S_3=1110\)
\(S_4=11110\)
\(\vdots\)
\(S_k=\sum_{k=0}^\infty10^k\rightarrow\infty\)
Now,
\(a_k=10^k,\ a_{k+1}=10^{k+1}\)
\(\lim_{k\rightarrow\infty}\frac{10^k}{10^{k+1}}=\lim\frac1{10}=\frac{1}{10}<1\)</span>
Hence the series is divergent.
0

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