## anonymous 4 years ago how to take the derivative of series? for example ((-1)^n+1)(x-2)^n)/n

1. anonymous

but you are a higher level than zarkon haha.

2. anonymous

hell no term by term

3. anonymous

is (-1)^n+1(x-2)^n-1 right?

4. anonymous

yes i believe so , let me write it so i don't do anything silly are we starting at n = 0 or n = 1?

5. anonymous

yes that is right. what you have

6. anonymous

1. okay and then i got the interval of that to be 1<x<3, but i am having trouble with the convergence of course.

7. anonymous

same

8. anonymous

oh but not at the endpoints of course

9. anonymous

next question...can you explain how to find the power series of a function?

10. anonymous

i can show you a trick for that one if you like

11. anonymous

well maybe not

12. anonymous

yes please! i have an exam tomorrow and i need as much help as possible.

13. anonymous

well then i don't want to waste your time with this. but if you know that $\frac{1}{1-x}=\sum x^n$ then $\frac{1}{1+x}=\sum (-1)^nx^n$ and so $\frac{1}{x-1}=\frac{1}{1+(x-2)}=\sum(-1)^n(x-2)^n$ so that is your derivative, and therefore your original series was the log

14. anonymous

but if you have an exam forget that mess

15. anonymous

and also your derivative will not convege at the endpoints, because there is not way that $\sum 1^n$ or $\sum (-1)^n$ converges

16. anonymous

this is not wasting my time at alll. umm soo what about htis: use the binomial series to find the maclaurin series of the function f(x) = $\sqrt[4]{1+x}$

17. anonymous

i guess you are supposed to write this as $(1+x)^{\frac{1}{4}}$ first and then use "general binomial series

18. anonymous

er $1+\frac{1}{4}x+\frac{\frac{1}{4}-1}{2!}x^2$

19. anonymous

$+\frac{(\frac{1}{4}-1)(\frac{1}{4}-2)}{3!}x^3 +...$

20. anonymous

probably some algebra will turn up a nice pattern

21. anonymous

but i dont understand the maclaurin part?

22. anonymous

that is the maclaurin series

23. anonymous

expand about zero, get $a_0+a_1x+a_2x^2+a_3x^3 + ...$

24. anonymous

you can do this using the usual derivative method, but the generalized binomial formula will work in this case

25. anonymous

wait what about the thing where you take the derivative multiple times and plug it in?

26. anonymous

yes you can do that, but it is a pain. problem said "binomial" so i used it. take a look here http://www.proofwiki.org/wiki/Binomial_Theorem/General_Binomial_Theorem

27. anonymous

the hint was "binomial" formula

28. anonymous

ohh oops. i am terrible at math! and context clues i guess haha. well, do you have any test-taking tips ?

29. anonymous

get some sleep and don't study right before the test, it will only freak you out when you see what you don't know ok easier said than done, i know relax though, it helps

30. anonymous

but i do not believe you are terrible at math because this is fairly advanced stuff, and in any case you found the intervals of convergence yourself also look at assigned homework problmes, and especialy quizzes because professors tend to repeat themselves, or at least ask the same types of questions

31. anonymous

good luck

32. anonymous

that first part of the tip is so true! anyway thank you so much. you are a lifesaver. seriously.

33. anonymous

your quite welcome, and really good luck and relax and don't stay up all night studying