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

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

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sum from n=1 to infinity of 1/(n-1)-1/n

n = 1 \sum \infty \frac{1}{n - 1} - \frac{1}{n}

1

Solution steps

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

Apply Telescoping Series Test:

1

= 1

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

n = 2 \sum \infty (\frac{5}{6})^{3 n}

n = 1 \sum \infty \frac{1 8 ^{n}}{n !}

k = 0 \sum \infty (\frac{k}{k + 15})^{k}

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