What are the rules for using the limit comparison theorem in power series?

[Richter915]

Hi, I'm preparing for my calculus II final and I had a question about Power Series. I'm posting the link for the answer solution to a practice exam (it's in pdf form) and I will ask questions based on that.

http://www.math.sunysb.edu/~daryl/prcsol3.pdf

On problem 12 it asks for the fifth derivative, I'm not sure why they did the last part. I can do the first three steps and solve for the power series of f(x) but I have no clue why they do that last part so if you can clear it up it would be appreciated.

On problem 7 part F...can you please explain the rules for limit comparison theorem. Also, when facing a problem like that, what series are good to use in comparison to the series given to you?

Thanks for any help, more questions will be asked in the future...heh.

 

[quasar987]

Hi, I'm preparing for my calculus II final and I had a question about Power Series. I'm posting the link for the answer solution to a practice exam (it's in pdf form) and I will ask questions based on that.

http://www.math.sunysb.edu/~daryl/prcsol3.pdf

On problem 12 it asks for the fifth derivative, I'm not sure why they did the last part. I can do the first three steps and solve for the power series of f(x) but I have no clue why they do that last part so if you can clear it up it would be appreciated..

They use the fact that if a function f can be written as a power serie $\sum a_kx^k$, then $a_k = f^{(k)}(0)/k!$. So they only look at the coefficient of x^5 and they're assured that it is $f^{5}(0)/5!$.

 

[quasar987]

On problem 7 part F...can you please explain the rules for limit comparison theorem. Also, when facing a problem like that, what series are good to use in comparison to the series given to you?

How it works: You chose a serie of positive n-th term b_n for which you know wheter it converges or not. Then you compute the limit

$$\lim_{n\rightarrow \infty}\frac{|a_n|}{b_n}$$

i) If the result is any number that is not 0 or infinity, the two series have the same convergence value. So if b_n diverges, a_n too and if b_n converges, a_n too.

ii) If the limit is 0, and if b_n converges, |a_n| converges.

iii) If the limit is $\infty$ and if b_n diverges, then |a_n| diverges.

Otherwise, you cannot conclude.

Good series to use as b_n are the "Riemman p-series":

$$\sum \frac{1}{n^p}$$

it diverges for $p \leq 1$ and converges for $p>1$.

 

[Richter915]

http://www.math.sunysb.edu/~leontak/prfinal.pdf

http://www.math.sunysb.edu/~leontak/solpf.pdf

Hi I was unsure of the explanation of the answer for question 13 part a. I understand everything up to where he finds the pattern shown by the recursive relation but after that, I'm not sure how he solves for y(x).

Also, can you please explain the solution to 14b. I don't understand the thing with the long division, everything before that I recognize..please help...thanks.

 

[Richter915]

How it works: You chose a serie of positive n-th term b_n for which you know wheter it converges or not. Then you compute the limit

$$\lim_{n\rightarrow \infty}\frac{|a_n|}{b_n}$$

i) If the result is any number that is not 0 or infinity, the two series have the same convergence value. So if b_n diverges, a_n too and if b_n converges, a_n too.

ii) If the limit is 0, and if b_n converges, |a_n| converges.

iii) If the limit is $\infty$ and if b_n diverges, then |a_n| diverges.

Otherwise, you cannot conclude.

Good series to use as b_n are the "Riemman p-series":

$$\sum \frac{1}{n^p}$$

it diverges for $p \leq 1$ and converges for $p>1$.

so with points ii and iii...what if b_n was divergent and the lim was 0? What would that mean? And with iii, what if the lim goes to infinity and b_n converges? what happens then?

Thanks.

 

[quasar987]

so with points ii and iii...what if b_n was divergent and the lim was 0? What would that mean? And with iii, what if the lim goes to infinity and b_n converges? what happens then?

These cases are covered in "Otherwise, you cannot conclude."

That is to say, if we get that b_n was divergent and the lim was 0 or that the lim goes to infinity and b_n converges, the test doesn't tell us anything about the convergence of |a_n|, and we have to try comparing it with another b_n or try another test.