# Sum of two infinite series: Σ[1/(36r^2-1)+2/(36r^2-1)^2]

• MHB
• juantheron
In summary, the sum of two infinite series can be calculated using the formula Σ(a + b) = Σa + Σb, where a and b are two infinite series. To find the sum, we need to first calculate the individual sums of each series and then combine them. However, the sum may not always be convergent as it depends on the convergence properties of the individual series. To determine if the sum is convergent or divergent, various convergence tests can be used. Additionally, the sum of two infinite series can be negative or positive depending on the values of the individual series.
juantheron
Evaluation of $$\displaystyle \displaystyle \sum_{r=1}^\infty \left(\frac{1}{36r^2-1}+\frac{2}{(36r^2-1)^2}\right)$$

In partial fractions,
$$\frac{1}{36r^2-1}+\frac{2}{(36r^2-1)^2}\ =\ \frac12\left[\frac1{(6r-1)^2}+\frac1{(6r+1)^2}\right].$$
So the sum is equal to
$$\frac12\left(\frac1{5^2}+\frac1{7^2}+\frac1{11^2}+\frac1{13^2}+\cdots\right)$$

$=\ \frac12(A-B-C+D)-\frac12$

where

$\displaystyle A\ =\ \frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\cdots\ =\ \frac{\pi^2}6$,

$\displaystyle B\ =\ \frac1{2^2}+\frac1{4^2}+\frac1{6^2}+\cdots\ =\ \frac14\cdot\frac{\pi^2}6$,

$\displaystyle C\ =\ \frac1{3^2}+\frac1{6^2}+\frac1{9^2}+\cdots\ =\ \frac19\cdot\frac{\pi^2}6$,

$\displaystyle D\ =\ \frac1{6^2}+\frac1{12^2}+\frac1{18^2}+\cdots\ =\ \frac1{36}\cdot\frac{\pi^2}6$.

Hence:
$$\sum_{r=1}^\infty\left[\frac1{36r^2-1}+\frac2{(36r^2-1)^2}\right]\ =\ \frac12\left(1-\frac14-\frac19+\frac1{36}\right)\frac{\pi^2}6-\frac12\ =\ \boxed{\frac{\pi^2}{18}-\frac12}.$$

Thanks https://mathhelpboards.com/members/olinguito/. My solution is almost same as yours.

## 1. What is the formula for the sum of two infinite series?

The formula for the sum of two infinite series is given by Σ(a + b) = Σa + Σb, where a and b are two infinite series.

## 2. How do you calculate the sum of two infinite series?

To calculate the sum of two infinite series, we need to first find the individual sums of each series using their respective formulas. Then, we can add the individual sums together to get the final sum of the two infinite series.

## 3. Is the sum of two infinite series always convergent?

No, the sum of two infinite series may not always be convergent. It depends on the individual series and their convergence properties. If both series are convergent, then the sum will also be convergent. However, if one or both series are divergent, then the sum will also be divergent.

## 4. How can we determine if the sum of two infinite series is convergent or divergent?

The convergence or divergence of a series can be determined by using various convergence tests such as the comparison test, ratio test, or root test. These tests can help us determine the behavior of a series and whether it will converge or diverge.

## 5. Can the sum of two infinite series be negative?

Yes, the sum of two infinite series can be negative. It depends on the individual series and their values. If both series have negative terms, then the sum will also be negative. However, if one or both series have positive terms, then the sum can be positive or negative depending on their values.

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