Question

7. Determine the radius of convergence and the interval of convergence for the power series: (-1)"+1" n2 nxn b. An=1 > C. X- + 3! 5! 8. A probability density function f(x) is an important concept in statistical sciences. It gives you the distribution of the random variable x. f(x) usually defined in a certain interval, and vanish in the rest. One can defined the median j and variances oas using the probability density function as (you'll see more about this later on in the course of statistic): u=1*xf(x)dx (x - x) dx For most cases the distribution function is normal or Gaussian. If my distribution has the form of exponential distribution: f(x) = Şbe-bx, aear, X>0 x<0 determine the median u and the variances o2 according to the previous formula!

Answer 1

(n+1) 2 (-)ht ht n x § (-1) na1 h2 ht и (ht) + nel (-o) (92 2 z n so (-D an anti nz (na) 2 n+2 (1) x (2) x n² antl n lim 2 XN an nD nya 2 at (x in X lim X(-1) xn dxn (n+1)? x (-y+41 ху n D (-1 xa 2 lim nD (n+12 ha lim nts -N 2. - N N lim naD ["46] l-nl (1 +424 1614_]] =0) => l-nl a tale lal | --H1x1n/= tal z 2 (140)

for any secies to be convergent anal < 1 lim nad an here so n ranges so hence lal < from - ll lnk 1 for (al should be les than 1. range interval of convergence -kna 1 zadines 1 of convergence Since Inlk 1 the ri right hand side is the radius of convergence 7 и nn b. E n21 ntl n 2 (n+1) a an anal 2 nn ht antl lim (nt a 2 lim an nto и ЭА nnn 2 lim хи nto

lim nyo (44) ** (1+)xtol Txalami since / the limit came same as the previous question the Guswer is same as the previous question. interval -1 <n< لے relius of convergence z 1 < 1 => [nl<1 Since Photos tales > 101<3) 7. C. 2- N + ulü 3! 7! 3 a tas 51 ntl "! 3! 7! (2n-1) So (-1) x (2) (2n-1)! You can put na 1, 2, 3 ets to get respective terms

heme a 2 + n-nt .. 5! 71 I ! 3 b ht 2n-1 (-1) X (2) (2n-1)! na1 ntl anal (helt 2(nt.) (1) x^2 an 2 (-1) () 2 anal (2n-1)! [2 cm +1) - 13 ! ht2 anta-1 (-1) X (n) anti lim a lim an n D и >> (@n+2-1)! X (2n-1)! Qut (-1) + x(n) (-1) x (m) Intl nta lim Z (2n-1)! (-1) htlx (2) 2nd nye (an+0!

اره 2 lim x nx (2) (-1) X (-17 x (2nd)! (2n +1) (Qu) (2,4) ! Au course 2 (-1) xa? lim z no (2n+1) (and 2 lim |(-131 · 1 => nto Qu) (anti) 2 2 ㅋ = N. N. ola oll (2) (2db +0) which is less than '1' (0x2] regardles of the value of (n), unless We find in the previous two examples hence is conreegent for any value of a ) (-9 <n<os), the given selies convergent. Now it oscillates between hence the radius of Convergence ۱) (هه ده - (R=0 /

8. fine 26 bn Фе a7o. an a e ato M z In anda u z Sugoden & Sagada / because the function has two values [bet Ots, and a can for no, an GY nyo, / for a -9->) an Suxae x x ae de of Sexbe M. - bn axbe da Sa nende toga grebenda ILATE = algebric comes before exporential hence become the first function.

an ne -bn ne upua + b gr du 2 W e up da a 2 up [ax Sean Sox[se"aw) dan Six(se*3.) din Suv. in ho-Slow x su + b [n Setan accosting to this rule an e M e X u a se) dan a -bn -bu nxe - b dn + b - b -neba M 2 an Ixe a ne + b 0 b -bn + lie b -b

-bn an ne b Me -bn -ne - le b + ti Ecco 62 az axa ax-00 - 8 хе ax-D axo oxe + lxe fle ма a 92 9 G2 -bxo -bxd gre + b b be -bxo - bxo Ixe oxe 2 b b M |60**"). [+ (-0- of b (-0-0) - 1 X 1 62 6xo) blo) ao – be e e Since e b u z + a ( Vair) 1 + 1 9 b