Question

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

Short answer

The sum of$1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}$can be written in the summation notation as$\sum _{n=1}^5\frac{1}{n^2}.$

Step-by-step solution

Step 1. Given information

$\begin{array}{l}\text{Thegivensumis}\\ 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}\\ \text{Wehavetowritethesumusingsummationnotation.}\end{array}$

Step 2. Find the pattern

Now,

We look for a pattern in the terms.

$\begin{array}{cccccccc}n:& 1& 2& 3& 4& 5& \mathrm{...}& n\\ \text{Terms}& \frac{1}{1}& \frac{1}{4}& \frac{1}{9}& \frac{1}{16}& \frac{1}{25}& \mathrm{...}& \\ \text{Pattern}& \frac{1}{1^2}& \frac{1}{2^2}& \frac{1}{3^2}& \frac{1}{4^2}& \frac{1}{5^2}& \mathrm{...}& \frac{1}{n^2}\end{array}$

Step 3. Conclusion

From the above table, we can observe that all the numerators are 1 and the denominators are power of the term.

Therefore,

We can write the sum for the 5 terms in summation notation as

$\sum _{n=1}^5\frac{1}{n^2}.$