Solve sum(1/n-1/n+1)

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

Evaluate

$$\frac{Σ}{n\left(n+1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of n and n+1 is n\left(n+1\right). Multiply \frac{1}{n} times \frac{n+1}{n+1}. Multiply \frac{1}{n+1} times \frac{n}{n}.

$$Σ\left(\frac{n+1}{n\left(n+1\right)}-\frac{n}{n\left(n+1\right)}\right)$$

Since \frac{n+1}{n\left(n+1\right)} and \frac{n}{n\left(n+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in n+1-n.

$$Σ\times \frac{1}{n\left(n+1\right)}$$

Express Σ\times \frac{1}{n\left(n+1\right)} as a single fraction.

$$\frac{Σ}{n\left(n+1\right)}$$

Use the distributive property to multiply n by n+1.

$$\frac{Σ}{n^{2}+n}$$

Expand

$$\frac{Σ}{n\left(n+1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of n and n+1 is n\left(n+1\right). Multiply \frac{1}{n} times \frac{n+1}{n+1}. Multiply \frac{1}{n+1} times \frac{n}{n}.

$$Σ\left(\frac{n+1}{n\left(n+1\right)}-\frac{n}{n\left(n+1\right)}\right)$$

Since \frac{n+1}{n\left(n+1\right)} and \frac{n}{n\left(n+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in n+1-n.

$$Σ\times \frac{1}{n\left(n+1\right)}$$

Express Σ\times \frac{1}{n\left(n+1\right)} as a single fraction.

$$\frac{Σ}{n\left(n+1\right)}$$

Use the distributive property to multiply n by n+1.

$$\frac{Σ}{n^{2}+n}$$