수열의 극한 전체 문제

In the following exercises, compute the general term ${a}_{n}$ of the series with the given partial sum ${S}_{n}.$ If the sequence of partial sums converges, find its limit $S.$

${S}_{n}=2-(n+2)/{2}^{n},n\ge 1$

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges.

$\sum_{n=1}^{\infty }\frac{n}{n+2}$

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges.

$\sum_{n=1}^{\infty }(1-{(-1)}^{n}))$

$\sum_{n=1}^{\infty }\frac{1}{(n+1)(n+2)}$ (Hint: Use a partial fraction decomposition like that for $\sum_{n=1}^{\infty }\frac{1}{n(n+1)}.)$

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges.

$\sum_{n=1}^{\infty }\frac{1}{2n+1}$ (Hint: Follow the reasoning for $\sum_{n=1}^{\infty }\frac{1}{n}.)$

Suppose that $\sum_{n=1}^{\infty }{a}_{n}=1,$ that $\sum_{n=1}^{\infty }{b}_{n}=-1,$ that ${a}_{1}=2,$ and ${b}_{1}=-3.$ Find the sum of the indicated series.

$\sum_{n=1}^{\infty }({a}_{n}+{b}_{n})$

Suppose that $\sum_{n=1}^{\infty }{a}_{n}=1,$ that $\sum_{n=1}^{\infty }{b}_{n}=-1,$ that ${a}_{1}=2,$ and ${b}_{1}=-3.$ Find the sum of the indicated series.

$\sum_{n=1}^{\infty }({a}_{n}-2{b}_{n})$

Suppose that $\sum_{n=1}^{\infty }{a}_{n}=1,$ that $\sum_{n=1}^{\infty }{b}_{n}=-1,$ that ${a}_{1}=2,$ and ${b}_{1}=-3.$ Find the sum of the indicated series.

$\sum_{n=2}^{\infty }({a}_{n}-{b}_{n})$

Suppose that $\sum_{n=1}^{\infty }{a}_{n}=1,$ that $\sum_{n=1}^{\infty }{b}_{n}=-1,$ that ${a}_{1}=2,$ and ${b}_{1}=-3.$ Find the sum of the indicated series.

$\sum_{n=1}^{\infty }(3{a}_{n+1}-4{b}_{n+1})$

State whether the given series converges and explain why.

$\sum_{n=1}^{\infty }\frac{1}{n+{10}^{80}}$ (Hint: Rewrite using a change of index.)

State whether the given series converges and explain why.

$1+\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots$

State whether the given series converges and explain why.

$1+\frac{e}{\pi }+\frac{{e}^{2}}{{\pi }^{2}}+\frac{{e}^{3}}{{\pi }^{3}}+\cdots$

State whether the given series converges and explain why.

$1-\sqrt{\frac{\pi }{3}}+\sqrt{\frac{{\pi }^{2}}{9}}-\sqrt{\frac{{\pi }^{3}}{27}}+\cdots$

For ${a}_{n}$ as follows, write the sum as a geometric series of the form $\sum_{n=1}^{\infty }a{r}^{n}.$ State whether the series converges and if it does, find the value of $\sum {a}_{n}.$

${a}_{1}=-1$ and ${a}_{n}/{a}_{n+1}=-5$ for $n\ge 1.$

For ${a}_{n}$ as follows, write the sum as a geometric series of the form $\sum_{n=1}^{\infty }a{r}^{n}.$ State whether the series converges and if it does, find the value of $\sum {a}_{n}.$

${a}_{1}=2$ and ${a}_{n}/{a}_{n+1}=1/2$ for $n\ge 1.$

For ${a}_{n}$ as follows, write the sum as a geometric series of the form $\sum_{n=1}^{\infty }a{r}^{n}.$ State whether the series converges and if it does, find the value of $\sum {a}_{n}.$

${a}_{1}=10$ and ${a}_{n}/{a}_{n+1}=10$ for $n\ge 1.$

For ${a}_{n}$ as follows, write the sum as a geometric series of the form $\sum_{n=1}^{\infty }a{r}^{n}.$ State whether the series converges and if it does, find the value of $\sum {a}_{n}.$

${a}_{1}=1/10$ and ${a}_{n}/{a}_{n+1}=-10$ for $n\ge 1.$

Use the identity $\frac{1}{1-y}=\sum_{n=0}^{\infty }{y}^{n}$ to express the function as a geometric series in the indicated term.

$\frac{x}{1+x}$ in $x$

Use the identity $\frac{1}{1-y}=\sum_{n=0}^{\infty }{y}^{n}$ to express the function as a geometric series in the indicated term.

$\frac{\sqrt{x}}{1-{x}^{3/2}}$ in $\sqrt{x}$

Use the identity $\frac{1}{1-y}=\sum_{n=0}^{\infty }{y}^{n}$ to express the function as a geometric series in the indicated term.

$\frac{1}{1+{\sin }^{2}x}$ in $\sin x$