Solve 5 sum n = 1 to n 1/((2n - 1)(2n + 1)) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\sum \frac { 1 } { ( 2 n - 1 ) ( 2 n + 1 ) }$$

Evaluate

$\frac{Σ}{4n^{2}-1}$
$|n|\neq \frac{1}{2}$

Short Solution Steps

Consider $\left(2n-1\right)\left(2n+1\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$Σ\times \frac{1}{\left(2n\right)^{2}-1^{2}}$$

Expand $\left(2n\right)^{2}$.

$$Σ\times \frac{1}{2^{2}n^{2}-1^{2}}$$

Calculate $2$ to the power of $2$ and get $4$.

$$Σ\times \frac{1}{4n^{2}-1^{2}}$$

Calculate $1$ to the power of $2$ and get $1$.

$$Σ\times \frac{1}{4n^{2}-1}$$

Express $Σ\times \frac{1}{4n^{2}-1}$ as a single fraction.

$$\frac{Σ}{4n^{2}-1}$$

Expand

$\frac{Σ}{4n^{2}-1}$
$|n|\neq \frac{1}{2}$

Short Solution Steps

Consider $\left(2n-1\right)\left(2n+1\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$Σ\times \frac{1}{\left(2n\right)^{2}-1^{2}}$$

Expand $\left(2n\right)^{2}$.

$$Σ\times \frac{1}{2^{2}n^{2}-1^{2}}$$

Calculate $2$ to the power of $2$ and get $4$.

$$Σ\times \frac{1}{4n^{2}-1^{2}}$$

Calculate $1$ to the power of $2$ and get $1$.

$$Σ\times \frac{1}{4n^{2}-1}$$

Express $Σ\times \frac{1}{4n^{2}-1}$ as a single fraction.

$$\frac{Σ}{4n^{2}-1}$$