Which series in Exercises 53–76 converge, and which diverge? G...

Question

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.

∑ (from n = 1 to ∞) (1 − 1/n)ⁿ

Accepted Answer

Welcome back everyone. Does the series sigma from n equals 1 up to infinity of 1 minus 2 divided by n to the power of n converge or diverge? If it converges, compute its sum. For this problem, we are going to use the test for divergence, and specifically we want to evaluate the limit as n approaches infinity of the nth term. In this case, that be limit as n approaches infinity of 1 minus 2 divided by n to the power of n. If this limit is not equal to zero, using the n term test, we can conclude that the series diverges. Otherwise, the test is inconclusive. So let's remember that the limit as n approaches positive or negative infinity of 1 + 1 divided by n to the power of n is equal to e. And this is how we will evaluate the limit as n approaches infinity of 1 minus 2 divided by n to the power of n. What we're going to do is adjust our term -2 divided by m in the form of 1 divided by n. To do that, we will just change the variable. So let's suppose that -2 divided by n is equal to 1 divided by, let's say, k. And now let's solve for the exponent and using cross multiplication, and multiplied by 1 is equal to -2 multiplied by k. And now as n approaches infinity. We can show that k, which is negative and divided by 2, while it is going to approach. In negative infinity divided by 2, which is negative infinity, so our limit is going to be a limit as k approaches negative infinity of 1. Plus 1 divided by K, race 2-2K. And using the properties of exponents we can just leave. Positivek as the exponent and then we're going to raise the whole limit to the power of -2. So this limit is equal to e raise the power of -2, which is equal to 1 divided by e2, and it is not equal to 0 because it is not equal to 0. The series diverges. And this is our final answer for this problem. Thank you for watching.

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.∑ (from n = 1 to ∞) (1 − 1/n)ⁿ

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Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 1 to ∞) (1 − 1/n)ⁿ