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Prove the formula for the sum of an infinite geometric series.

The formula for the sum of an infinite geometric series is S = a/(1-r), where a is the first term and r is the common ratio.

To prove this formula, we start with the formula for the sum of a finite geometric series:

S_n = a(1-r^n)/(1-r)

where S_n is the sum of the first n terms of the series. We want to find the sum of the infinite series, so we take the limit as n approaches infinity:

lim S_n = lim a(1-r^n)/(1-r)

As n approaches infinity, r^n approaches zero if |r|<1, so we can simplify the expression:

lim S_n = a/(1-r)

Therefore, the sum of an infinite geometric series is S = a/(1-r), as required.

We can also prove this formula using the formula for the sum of an infinite series:

S = a/(1-r) + ar/(1-r)^2 + ar^2/(1-r)^3 + ...

Multiplying both sides by (1-r), we get:

S(1-r) = a + ar/(1-r) + ar^2/(1-r)^2 + ...

Subtracting the second equation from the first, we get:

S - Sr = a

Solving for S, we get:

S = a/(1-r)

This is the same formula we obtained earlier. Therefore, we have proved the formula for the sum of an infinite geometric series.

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