Question

I need help with problem #9

For Questions 5 through 7, determine convergence or divergence of the series using any test of your choice. If the series is geometric, give its sum. You must verify that all assumptions of the test you choose are satisfied. 5. (8 points) § (193) 6. (10 points) { $s 8. Find the radius of convergence and interval of convergence for the series below: (a) (6 points) mal (b) (6 points) { }(22+1)* sin 9. (6 points) Show that the series 2 is absolutely convergent. - n=1 n!

Answer 1

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3nt) lim mm (3"-1) 3 Linn 3" 32 (-1) 3 M-94 linn m - - lining Chalon) sinó nt) i Simce buis comit exists and is converges componision test ... from thee limil & Gn't also converges n=1 32 and Find internal of the of convergence Radius convergence solution Given that antl. 22.0771) anti : 2 (nt) 2+2 (n+1)! ant) : from Radio test nu

La lim 1 2252 m! munbe l momentel ~ Lloyd y L-021 for Then the series is convergent any e value. - - Then Radius ab convergence Rsé C-2, 2) Inder vall at convergence solution . Given that nan [an] (n+1) anti - MAI . anti mth Centiti) from the Radio test anti Liliane on am me

- Le lim 1m 4-1) 21 in E bin 1 (nap) ant) (not) I m--> lenti( cmtv'no' - mun | Betten m-se / m2 m+ (1+infrum') .en wollen 1. =) La linn INCItyn) 7.4 - La linn | [+lm) x m elz citron] Anon) (tl) 1 it's z ci) =L: 1 (1+0)2.0)) a La 1² 시들시 -2< x 22 Now we check x=8-2 n 20(n²+1) =) Ĉ Then it is mal 2ncn²+1) Diverges

pow put x-2 m (2) n=1 Incr²t1) - Then it is also Diverges Now is the Padius at convergence R=2-(-2) 2 R = 2 Interval at convergence is (-2,2) Thank you,