What is the sum from k=1 to infinity of 2/k-2/(k+1) ?

The sum from k=1 to infinity of 2/k-2/(k+1) is 2

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sum from k=1 to infinity of 2/k-2/(k+1)

k = 1 \sum \infty \frac{2}{k} - \frac{2}{k + 1}

2

Solution steps

k = 1 \sum \infty \frac{2}{k} - \frac{2}{k + 1}

Expand

\frac{2}{k} - \frac{2}{k + 1} : \frac{2}{k ^{2} + k}

= k = 1 \sum \infty \frac{2}{k ^{2} + k}

Apply the constant multiplication rule:

\sum c \cdot a_{n} = c \cdot \sum a_{n}

= 2 \cdot k = 1 \sum \infty \frac{1}{k ^{2} + k}

Apply Telescoping Series Test:

1

= 2 \cdot 1

Simplify

= 2

What is the sum from k=1 to infinity of 2/k-2/(k+1) ?
The sum from k=1 to infinity of 2/k-2/(k+1) is 2