# 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
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.
$$\sum_{k=1}^\infty10^k$$

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.

### Relevant Questions

Consider the following infinite series.
a.Find the first four partial sums $$S_1,S_2,S_3,$$ and $$S_4$$ of the series.
b.Find a formula for the nth partial sum $$S_n$$ of the indinite series.Use this formula to find the next four partial sums $$S_5,S_6,S_7$$ and $$S_8$$ of the infinite series.
c.Make a conjecture for the value of the series.
$$\sum_{k=1}^\infty\frac{2}{(2k-1)(2k+1)}$$
A) Does this series converge? If yes, towards what number?
B) Find the first 5 terms of the sequence of partial sums in this series.
C) What is the general term of this sequence of partial sums?
$$\sum_{k=1}^\infty(\frac{2}{k}-\frac{2}{k+1})$$
Evaluating series. Evaluate the following infinite series or state that the series diverges.
$$\sum_{k=1}^\infty\frac{9}{(3k-2)(3k+1)}$$
Evaluate the following infinite series or state that the series diverges.
$$\sum_{k=1}^\infty4^{-3k}$$
Evaluating series. Evaluate the following infinite series or state that the series diverges.
$$\displaystyle{\sum_{{{k}={1}}}^{\infty}}{\frac{{{9}}}{{{\left({3}{k}-{2}\right)}{\left({3}{k}+{1}\right)}}}}$$
Consider the telescoping series
$$\sum_{k=3}^\infty(\sqrt{k}-\sqrt{k-2})$$
a) Apply the Divvergence Test to the series. Show all details. What conclusions, if any, can you make about the series?
b) Write out the partial sums $$s_3,s_4,s_5$$ and $$s_6$$.
c) Compute the n-th partial sum $$s_n$$, and put it in closed form.
Consider the telescoping series
$$\displaystyle{\sum_{{{k}={3}}}^{\infty}}{\left(\sqrt{{{k}}}-\sqrt{{{k}-{2}}}\right)}$$
a) Apply the Divvergence Test to the series. Show all details. What conclusions, if any, can you make about the series?
b) Write out the partial sums $$\displaystyle{s}_{{3}},{s}_{{4}},{s}_{{5}}$$ and $$\displaystyle{s}_{{6}}$$.
c) Compute the n-th partial sum $$\displaystyle{s}_{{n}}$$, and put it in closed form.
$$f(x)=\frac{1}{\sqrt{x}}$$ with $$a=4$$, approximate $$\frac{1}{\sqrt{3}}$$
$$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{\sqrt{{{x}}}}}}$$ with $$\displaystyle{a}={4}$$, approximate $$\displaystyle{\frac{{{1}}}{{\sqrt{{{3}}}}}}$$
b. Find how many terms are needed to ensure that the remainder is less than $$10^{-3}$$.
$$\sum_{k=1}^\infty\frac{1}{3^k}$$