Question

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that R_n(x) → 0.] Find the associated radius of convergence, R.

f(x) = e^−2x

Please show all work, and explain it in great detail. Please be sure to include all rules, theorems, and algebra (even if you are performing simple multiplication, please still show it). Please do not use cursive. I am having a difficult time trying to grasp series, therefore, please adhere to the above requests. Thank you in advance for your help!!!

Answer 1

cele Taylor series expansion of any function for around (a=o) point is known cas macdaurin series so macclausin series of a given function for about the point a=0 is given as for – to tato + x2 fő ta3 flog 21 21 31 -- sinfo ---- n! - © Now we have to find the macadaron Series expansion of for = ė 2x $0 we calculate some forms accor- -ding to ® - then so = 2 20 = e = 1 fon = & 2* (-2) : - 2 e 2x ② E - ě 2800 - - 2 x 1 - -2 Again differentiate o the then nur quill get a J CI

- - 2 -24 € (-2) -22 é - + q dow to = 4 e 2(0) then fo = -8 Now differentiafe & again then f = 4 e 24 (-2) = -8 € 21 - fro = 8 e 2(0) 8x2 = -2 Similarly we can calculate other terms as well. yow put the valeces of food, fue flo ... in D those cue will 9 ми с Сости отремо, fo= ė 2x a 2x € 1-20 +422 - 823 + M 21 31 aftor observing the series wee can get other terens early

. 2x 1 = t 16 ng 1- 2x + 9 2² – 243 -t n - 8 n LI 21 31 4! ml Lle coln sexies ------ + (-2)^2 + --- then we can write aboue series e * = Ž -2)^2 -> en- on re (min stand for factorial (nl= non la n.2. 13:21 ylow we will caccla late the radices of convergence of B we know Roedius of convergence (R) of a series to like an an M20 for Canah. - © 80 an= (-22h, if une comperse and © . lule ACO и

then aut = (-2) ht nel flow auts - (~2) ht nell an (-2)2 ul - (2)4+1 (-2) (nt) ! (-234 m xr2t4+1-4) (461) W = -22+ then url си Иt) antl yam l hao - ling a ontl 2 X Ce nzo Hence feme" Infinite radius of ), en tres convergence.
hey dear this is your complete solution in detail. Plz like it. ???Thank you.