# How do you find the nth partial sum, determine whether the series converges and find the sum when it exists given #1+3/4+9/16+...+(3/4)^n+...#?

To find the nth partial sum of the series (1 + \frac{3}{4} + \frac{9}{16} + \ldots + \left(\frac{3}{4}\right)^n + \ldots), you use the formula for the sum of a geometric series. The formula is (S_n = \frac{a(1 - r^n)}{1 - r}), where (S_n) is the nth partial sum, (a) is the first term, (r) is the common ratio, and (n) is the number of terms.

For this series, the first term (a) is 1, and the common ratio (r) is ( \frac{3}{4} ). Plug these values into the formula to find the nth partial sum.

To determine whether the series converges, you check if the absolute value of the common ratio (r) is less than 1. If (|r| < 1), the series converges; otherwise, it diverges.

If the series converges, you use the formula for the sum of an infinite geometric series, which is (S = \frac{a}{1 - r}), where (S) is the sum, (a) is the first term, and (r) is the common ratio.

Plug in the values of (a) and (r) into the formula to find the sum of the series if it exists.

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Consider the series:

and its partial sum:

and:

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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