Question

Write the following using summation notation. $-1+\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}$

Short answer

The summation notation for the sequence $-1+\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}$ is $\sum _{n=1}^5\frac{{(-1)}^n}{n^2}$

Step-by-step solution

Step 1. Given

The expression $-1+\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}$

To write the expression using sum notation.

Step 2. Look for the pattern in the term

The number of terms are $n=1,2,3,4,5$and the pattern will be like,

$-\frac{1}{1^2},\frac{1}{2^2},-\frac{1}{3^2},\frac{1}{4^2},-\frac{1}{5^2}$

Step 3. Write with summation notation

From the pattern, the numerator is the alternative negative sign while the denominator is the square of numbers from $1$to $5$.

So the summation notation is,

$-1+\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}=\sum _{n=1}^5\frac{{(-1)}^n}{n^2}$