# How to calculate this limit?#lim_(n->oo)sum_(k=1)^n(1/2^k+1/3^k)#

The general term of a geometric series can be written:

Then we find:

So we find:

Then:

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To calculate the limit lim_(n->∞) sum_(k=1)^n (1/2^k + 1/3^k), we can first simplify the sum using the formula for the sum of a geometric series. The sum of the series 1/2^k for k from 1 to n is (1/2) * (1 - 1/2^n)/(1 - 1/2) = 1 - 1/2^n, and similarly, the sum of the series 1/3^k for k from 1 to n is (1/3) * (1 - 1/3^n)/(1 - 1/3) = 1 - 1/3^n.

Therefore, the sum of the given series is (1 - 1/2^n) + (1 - 1/3^n), which simplifies to 2 - 1/2^n - 1/3^n. As n approaches infinity, both 1/2^n and 1/3^n approach 0, so the limit is 2 - 0 - 0 = 2.

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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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