How do you find the nth partial sum, determine whether the series converges and find the sum when it exists given #1+1/3+1/9+...+(1/3)^n+...#?
The given series is a geometric series with first term ( a = 1 ) and common ratio ( r = \frac{1}{3} ).

The nth partial sum of a geometric series is given by the formula: ( S_n = \frac{a(1  r^n)}{1  r} ).

To determine whether the series converges, recall the condition for convergence of a geometric series: It converges if and only if ( r < 1 ).

If ( r < 1 ), the sum of the geometric series is given by the formula for the sum of an infinite geometric series: ( S = \frac{a}{1  r} ).
Thus, for the given series:

The nth partial sum is ( S_n = \frac{1(1  (\frac{1}{3})^n)}{1  \frac{1}{3}} ).

The series converges since ( r = \frac{1}{3} < 1 ).

The sum of the series is ( S = \frac{1}{1  \frac{1}{3}} = \frac{3}{2} ).
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and note that:
so that:
And in general we can see that a geometric series converges when:
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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