Question

How do you use a calculator to find the sum of $\sum {\displaystyle \frac{{(-1)}^k}{k+1}}$ where $k$ is $[0,4]$?

Answer 1

Hint: Start by defining the summation. Then specify if there are any types. Then next start substituting values one by one in the summation term. After evaluating values of each of the terms, add all those values and evaluate the summation.

Complete step-by-step answer:
We will first start off by substituting the terms starting from $0$ to $4$. So, now we substitute $0$, and then evaluate the value of the term.
$$\begin{gathered} \Rightarrow {\displaystyle \frac{{(-1)}^k}{k+1}} \\ \Rightarrow {\displaystyle \frac{{(-1)}^0}{0+1}} \\ \Rightarrow {\displaystyle \frac{1}{1}} \\ \Rightarrow 1 \end{gathered}$$
So, now next we substitute $1$, and then evaluate the value of the term.
$$\begin{gathered} \Rightarrow {\displaystyle \frac{{(-1)}^k}{k+1}} \\ \Rightarrow {\displaystyle \frac{{(-1)}^1}{1+1}} \\ \Rightarrow {\displaystyle \frac{-1}{2}} \end{gathered}$$
So, now next we substitute $2$, and then evaluate the value of the term.
$$\begin{gathered} \Rightarrow {\displaystyle \frac{{(-1)}^k}{k+1}} \\ \Rightarrow {\displaystyle \frac{{(-1)}^2}{2+1}} \\ \Rightarrow {\displaystyle \frac{1}{3}} \end{gathered}$$
So, now next we substitute $3$, and then evaluate the value of the term.
$$\begin{gathered} \Rightarrow {\displaystyle \frac{{(-1)}^k}{k+1}} \\ \Rightarrow {\displaystyle \frac{{(-1)}^3}{3+1}} \\ \Rightarrow {\displaystyle \frac{-1}{4}} \end{gathered}$$
So, now next we substitute $4$, and then evaluate the value of the term.
$$\begin{gathered} \Rightarrow {\displaystyle \frac{{(-1)}^k}{k+1}} \\ \Rightarrow {\displaystyle \frac{{(-1)}^4}{4+1}} \\ \Rightarrow {\displaystyle \frac{1}{5}} \end{gathered}$$
Now that we have evaluated the value of each term by substitution of values, we will now add all the values in order to evaluate the summation.
$$\begin{gathered} \Rightarrow 1-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}-{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{5}} \\ \Rightarrow 0.783 \end{gathered}$$
Hence, the value of the summation $\sum {\displaystyle \frac{{(-1)}^k}{k+1}}$ is $0.783$.

Additional Information: Summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well like the functions, vectors, matrices, polynomials and in general, elements of any type of mathematical objects on which an operation.

Note: While substituting the terms, make sure you substitute along with their respective powers and signs. While adding the terms after evaluating the values make sure to add along with their signs, do not change their signs.