In Exercises 29–36 provide the first five terms of the sequence of partial sums for the given series. You may find it useful to refer to your answers to Exercises 21–28.

$\sum _{k=1}^{\infty }\frac{1}{k^2+2}$

$\sum _{k=1}^{\infty }\frac{1}{k^2+2}$

The first term of the series $\sum _{k=1}^{\infty }\frac{1}{k^2+2}$is obtained by substituting $k=1$in $\frac{1}{k^2+2}$. Therefore, the value at $k=1$ is:

$\frac{1}{(1{)}^2+2}=\frac{1}{1+2}$ (Substituting)

$=\frac{1}{3}$

The first term of the series $\sum _{k=1}^{\infty }\frac{1}{k^2+2}$is $\frac{1}{3}$.

The second term of the series $\sum _{k=1}^{\infty }\frac{1}{k^2+2}$is obtained by substituting $k=2$in $\frac{1}{k^2+2}$. Therefore, the value at $k=2$is:

$\frac{1}{(2{)}^2+2}=\frac{1}{4+2}$(Substituting)

$=\frac{1}{6}$

The second term of the series $\sum _{k=1}^{\infty }\frac{1}{k^2+2}$is $\frac{1}{6}$.

The third term of the series $\sum _{k=1}^{\infty }\frac{1}{k^2+2}$is obtained by substituting $k=3$in $\frac{1}{k^2+2}$. Therefore, the value at $k=3$ is:

$\frac{1}{(3{)}^2+2}=\frac{1}{9+2}$(Substituting)

$=\frac{1}{11}$

The third term of the series $\sum _{k=1}^{\infty }\frac{1}{k^2+2}$is $\frac{1}{11}$.

The fourth term of the series $\sum _{k=1}^{\infty }\frac{1}{k^2+2}$is obtained by substituting $k=4$in $\frac{1}{k^2+2}$. Therefore, the value at $k=4$ is:

$\frac{1}{(4{)}^2+2}=\frac{1}{16+2}$ (Substituting)

$=\frac{1}{18}$

The fourth term of the series $\sum _{k=1}^{\infty }\frac{1}{k^2+2}$is $\frac{1}{18}$.

The fifth term of the series $\sum _{k=1}^{\infty }\frac{1}{k^2+2}$is obtained by substituting $k=5$in $\frac{1}{k^2+2}$. Therefore, the value at $k=5$ is:

$\frac{1}{(5{)}^2+2}=\frac{1}{25+2}$ (Substituting)

$=\frac{1}{27}$

The fifth term of the series $\sum _{k=1}^{\infty }\frac{1}{k^2+2}$is $\frac{1}{27}$.

The first five terms in the sequence of partial sums are:

$$\begin{gathered} S_1=\frac{1}{3} \\ {S}_2=S_1+a_2 \\ =\frac{1}{3}+\frac{1}{6}\text{(Substitution)} \\ =\frac{1}{2} \\ {S}_3=S_2+a_3 \\ =\frac{1}{2}+\frac{1}{11}\text{(Substitution)} \\ =\frac{13}{22} \end{gathered}$$

$$\begin{gathered} S_4=S_3+a_4 \\ =\frac{13}{22}+\frac{1}{18}\text{(Substitution)} \\ =\frac{117+11}{198} \\ =\frac{64}{99} \\ {S}_5=S_4+a_5 \\ =\frac{1}{27}+\frac{64}{99}\text{(Substitution)} \\ =\frac{203}{297} \end{gathered}$$

Therefore, first five terms of partial sums for the given series is $\left\{\frac{1}{3},\frac{1}{2},\frac{13}{22},\frac{64}{99},\frac{203}{297}\right\}$.