# 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. Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.
$$f(x)=\frac{1}{\sqrt{x}}$$ 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 $$10^{-3}$$.
$$\sum_{k=1}^\infty\frac{1}{3^k}$$