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

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

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

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

1

Solution steps

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

Apply Telescoping Series Test:

1

= 1

n = 2 \sum \infty \frac{n !}{( 2 n ) !}

n = 1 \sum \infty \frac{1}{n ^{3} + n}

n = 1 \sum \infty \frac{1}{n ^{3} + 8}

n = 1 \sum \infty \frac{n}{n + 10}

n = 1 \sum \infty \frac{1}{n ^{5} + 2}

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