Series, limits: sum[n=3] [n/(n+1) - (n+2)/(n+3)], lim[x->0^+] [(e^{2sqrt(x)} - 1)/...

[lolasilves]

Series, limits: sum[n=3] [n/(n+1) - (n+2)/(n+3)], lim[x->0^+] [(e^{2sqrt(x)} - 1)/...

I have 2 exercicies that I can´t solve.

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1. Determinate the nature and, if possible, the sum:

. . .\(\displaystyle \displaystyle \sum_{n=3}\, \left(\dfrac{n}{n\, +\, 1}\, -\, \dfrac{n\, +\, 2}{n\, +\, 3}\right)\)

2. Calculate the limit:

. . .\(\displaystyle \displaystyle \lim_{x \rightarrow 0^+}\, \dfrac{e^{2\sqrt{x\,}}\, -\, 1}{\tan\big(\sqrt{ x\,}\big)}\)

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"lim_{x→(0^+)}" means to determinate the limit of 0 that tend from the positive numbers.

Can some one please help me to understand this?
I'm pretty sure she will put someting similar on the exam. Thank you!

 

[tkhunny]

For the first, what does "n = 3" mean? Is there an upper limit?

For the second, have you any tools? What have you been discussing in class? l'Hôpital?

 

[lolasilves]

For the first, what does "n = 3" mean? Is there an upper limit?

Fir the second, have you any tools? What have you been discussing in class? l'Hôpital?

"n = 3" means "n = 3" and there is no upper limit. I'm confuse too i ask my colleages but none knows how to do it., and they all have the same exercises writen in the same way.
and the second with have studying l'hopital but no one knows how to solve it.

 

[stapel]

From what you have posted in reply to a previous helper, I will guess that you mean that the sum begins at "n = 3" and "goes to infinity", so the summation is:

1. Determinate the nature and, if possible, the sum:

. . .\(\displaystyle \displaystyle \sum_{n=3}^{\infty}\, \left(\dfrac{n}{n\, +\, 1}\, -\, \dfrac{n\, +\, 2}{n\, +\, 3}\right)\)

What does your textbook mean by "the nature [of] the sum"? What "natures" have you been given, from which to choose? What have you tried so far with this summation? For instance, did you notice anything when you wrote out the first few terms of the summation?

2. Calculate the limit:

. . .\(\displaystyle \displaystyle \lim_{x \rightarrow 0^+}\, \dfrac{e^{2\sqrt{x\,}}\, -\, 1}{\tan\big(\sqrt{ x\,}\big)}\)

What identities have you been given for this, if any? What proved (in the textbook, or in previous homework questions) limits do you have, which might be applicable here (for use in the Squeeze Theorem, etc)? Have you learned about l'Hospital's Rule yet? (I was able to see something helpful when I used this Rule, and was able to confirm that I'd been on the right track.)

When you reply, please include a clear listing of your thoughts and efforts so far. Thank you!