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

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

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sum from x=1 to infinity of 1/(x(x+1))

x = 1 \sum \infty \frac{1}{x ( x + 1 )}

1

Solution steps

x = 1 \sum \infty \frac{1}{x ( x + 1 )}

Apply Telescoping Series Test:

1

= 1

n = 1 \sum \infty \frac{1}{( 2 n + 1 ) !}

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

n = 1 \sum \infty \frac{2}{7 ^{n + 1}}

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

n = 1 \sum \infty 4 ln (\frac{n}{n + 1})

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