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Numerade Educator



Problem 25 Easy Difficulty

Determine whether the series converges or diverges.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {e^n + 1}{ne^n + 1} $




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Video Transcript

let's determine whether the Siri's convergence or diverges no. So here. Let's use the Lim comparison test from the section in the textbook. So here let me call this an and then let me define B end to be e n over Andy End. And then I could cancel, too. You just get one over Ed, and the reason I wrote beyond this way is because I just look it. Anne and I look at it, the larger terms in the numerator as N goes to infinity and same thing on the denominator. And then here I just cancelled out the easier than power. Now let's look at the limit of an over Bian. Then if I divide by one over and that's the same thing, just multiplying by N. And then here I should keep writing that limit. So I forgot to write the limit. Let me write that in there, Lim Lim. Now, as we take that limit first, let's go in, and you could use low Patel's rule here if you want. But here, let's just divide top and bottom by Andy to the end. So we'LL divide here the whole fraction by this term, and we have one plus one over Ian after canceling one plus one over any end. And then as we take that limit, since either then goes to infinity this denominators going to infinity. So that goes to zero The same thing for that denominator. So we just have one plus zero over one plus zero, which is one we know that the sum of the beings diverge Diverges. This is just a harmonic series. Or you could even use the pizza's here. Pee test with P equals one because that's the power of the end over here on the formula for being, so it diverges for that value of P. On the other hand, we also know that the limit that we just computed satisfies this inequality here because we're just computer that was equal to one in any time. This limit is bigger than zero in less than infinity. You can use the Lim comparison test and this tells you that a sum of a N and bien both converge or bull diverge. And since we already know that being diverges Bye, lct by Lim comparison test some abbreviating this El Sisi. We know that the sum of the A end. So let's write that out formally either the N plus one and either then plus one also beverages, and that's our final answer.