Question

QUESTION 16 For what values of x does the series converge conditionally? (-1)"(x + 5)" w n n=1 OA. x= -6 B. x = – 4 x= -6, x= -5 D. x=-6, x = -4 O E. x= -5,x= -4

Answer 1

Here (40) (x+5) is, given series SWE n= n So, Here an= (1) (atsin n we Inow, 2 am converges absolutely it' nel Šlanl and converges converges conditionally in nel CO 을 an converges but I land diverges, nel n=1 Now, ansider, anti lim nos = n (-1)+1 (x+s, n+1 lim п-ср an (n+1) con (x+5) ท - (-1) (x+5).n lim ntcs reitlin) = 120+51 lim ) р-н + こ Irts I 1 tlo - 120+57 (1) Ixt5] lim htus ant an

Lilim n-co anti net = 12+51 Now, Ration San conege 13y test, cue know) n=1 absolutely ito lim ant 1 21 nco ain ie. 10+5lci =) song -1 < xts < 1 =) -15 < x < 1-5 -6 <0 <-4 So, SW (-un (x+son an = sha cenvesges nal ni n absolutely for TE (-4,-6) For x=-4, given series becomes, Mood * (-1) (-ion (otsin con (-4+5) nal nel n & Ms Ms u/ (gen cun n = 1)ก n nal

8W (-1) (x+5)n (1) nal n. nei an= which is alternating series with So, lim lim sl- an = nos می -۴ st ilim an 다 ص (h رکی دال nganti > ] nt 1 an anti :: { an] is decreasing sequence .: By Leibnite Leibnite test giren series cen verges absolutely for X=-4. for 26= -6 giren series becomes (x+5) (-1)" (-6+5)n رندانN co co n (-ion n. n n=1 (1) M (tin 11 n n =) Was Wa SWT SWE an = (-1) n naj 11 -15 n=1 which is p-series withi with P=1 and are know, p-series, no converges l€ 27 | n=1

po- so, and diverges if' PSI So, Fox x= -6 given series diverges. Das given series converges anditionally for 4 only and diverges fes 6. oprien (13)-4 is correct choice 2 = X --