series help

[dmsjr12]

We've been doing series questions in class for about two weeks now, and I'll admit, for the most part, I'm pretty lost on this homework assignment. I'm hoping to get some help with a couple problems and hopefully use them to help me with the rest of the homework. Thanks in advance.

#1) Use the ratio test
The sum of n=1 to infinity of the series ((6n+5)7^n)/9^(n+1)

Ok here's where I got to on this one:

Since we are using the ratio test it becomes:

((6(n+1)+5)7^(n+1))/9^(n+2) * 9^(n+1)/((6n+5)7^n)

This is where is gets messy and also where I think I'm making the mistake. I keep getting the answer to be 77/45 so divergent, but that isn't correct. Can anyone help me out here?

#2) (sqrt(n+2)*(n+4)!)/e^n+6
I have no idea where to even start with this one other than the same as what I did in #1 which is to simply add 1 to each n and multiply by the reciprocal of the initial series, so any insight would be appreciated.

I'm hoping to use these two to help me with the rest of the homework, so if there are any extra little tips or tricks you might know that would help me with this type of problem or just series in general, I would be delighted to hear them. Thanks

 

[dmsjr12]

oh sorry, for #2 it also is the sum from n=1 to infinity

 

[daon]

We've been doing series questions in class for about two weeks now, and I'll admit, for the most part, I'm pretty lost on this homework assignment. I'm hoping to get some help with a couple problems and hopefully use them to help me with the rest of the homework. Thanks in advance.

#1) Use the ratio test
The sum of n=1 to infinity of the series ((6n+5)7^n)/9^(n+1)

Ok here's where I got to on this one:

Since we are using the ratio test it becomes:

((6(n+1)+5)7^(n+1))/9^(n+2) * 9^(n+1)/((6n+5)7^n)

This is where is gets messy and also where I think I'm making the mistake. I keep getting the answer to be 77/45 so divergent, but that isn't correct. Can anyone help me out here?

#2) (sqrt(n+2)*(n+4)!)/e^n+6
I have no idea where to even start with this one other than the same as what I did in #1 which is to simply add 1 to each n and multiply by the reciprocal of the initial series, so any insight would be appreciated.

I'm hoping to use these two to help me with the rest of the homework, so if there are any extra little tips or tricks you might know that would help me with this type of problem or just series in general, I would be delighted to hear them. Thanks

\(\displaystyle \L\frac{(6(n+1)+5)7^{n+1}}{9^{n+2}} \cdot \frac{9^{n+1}}{(6n+5)7^n} = \frac{7}{9} \cdot \frac{6n+11}{6n+5}\)

The limit of which is \(\displaystyle \L\frac{7}{9} \cdot \( \frac{6}{6}\)=\frac{7}{9}\). (If you don't remember the rules for the limit of polynomial ratios, here you can mutiply top and bottom of the second fraction by 1/n to get the same result.)