Answer Key. ( 1) n (2x+3) n. n n=1. (2x+3) n. = lim. < 1 or 2x+3 < 4. ( 1) ( 1) 2n n

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1 Math Midterm Eam #3 December, 3 Answer Key. [5 Points] Find the Interval and Radius of Convergence for the following power series. Analyze carefully and with full justification. Use Ratio Test. L lim a n+ a n lim n n+ +3 n+ n+ n+ n +3 n n+ n n +3 n n+ n lim The Ratio Test gives convergence for when +3 That is < +3 < 7 < < 7 < < Endpoints: 7 The original series becomes n n n n+ n n Recall the harmonic p-series Check: lim n+ n lim n n n n n+ n n n n n+ lim +3 n+ +3 n n+ n+ < or +3 <. n 7 n +3 n+ n n n n+ n+ with p is divergent. + n n n n +3 n+ n 7+3 n n+ n which is finite and non-zero < <. Therefore, these two series share the same behavior, so our series is also divergent by LCT. n The original series becomes n +3 n n n+ n n+ n n n+ which n n n is convergent by the Alternating Series Test. b n n+ >. lim b n lim n+ 3. b n > b n+ since n+ > n+

2 Finally, Interval of Convergence I 7, ] with Radius of Convergence R.. [ Points] Find the MacLaurin series representation for each of the following functions. State the Radius of Convergence for each series. Your answer should be in sigma notation. a f n n 7 n n n 7 n n+ Here need 7 <, so R 7. b f 7 sin3 n n+ First, sin. Here R. n+! Then, sin3 n 3 n+ n+! n 3 n+ n+. R still. n+! Finally, 7 sin 3 7 n 3 n+ n+ n+! n 3 n+ n+8 n+! R still. 3. [5 Points] a Write the MacLaurin Series representation for f e. Then write out at least the first 6 terms. First, e n Here R. Then, e n n n. n n Finally, e n n+. e 3 + 5! 7 3! + 9! ! b Write out the definition of the MacLaurin Series for any function f. Then write out at least the first 6 terms. In general, the MacLaurin Series for any f is given as

3 f+f + f! + f 3! 3 + f! + f ! c Use the formulas in part a and part b together to determine the tenth derivative of f e evaluated at. That is, compute f. HINT: Do not compute out those derivatives manually. Match the formulas from a and b and eamine the coefficients. From part a we see all the terms are odd powered. So there will be NO term in the MacLaurin Series for e. Match coefficients of like degreed terms: f!. [5 Points] since there is no term f a Write the MacLaurin Series representation for f arctan. First, arctan n n+ n+. Net, arctan n n+ n+ Finally, arctan n n+ n+ n n+ n+ n n+3 n+. b Use the MacLaurin Series representation for f arctan from part a to Estimate arctan d with error less than. Justify in words that your error is indeed less than. arctan d n n+3 n+ d n n+ n+n estimate Note this is an alternating series. Use the Alternating Series Estimation Theorem ASET. If we approimate the actual sum with only the first term, the error from the actual sum will be at most the absolute value of the net term,. Here < as desired. 3

4 5. [5 Points] Find the sum for each of the following series. a n π n+ 6 n+ n+! n π n+ 6 π sin n+! 6 b n ln7 n ln7+ ln7! ln73 3! + ln7! +... e ln7 e ln7 7 c n π n+ 36 n n! π n π n π 36 n n! πcos π 3 6 d n n arctan π 6. [ Points] Volumes of Revolution a Consider the region bounded by y e +, y sin,, and π. Rotate this region about the horizontal line y. Set-up, BUT DO NOT EVALUATE!!, the integral to compute the volume of the resulting solid using the Washer Method. Sketch the solid, along with one of the approimating washers. See me for a sketch. V π π [ outer radius inner radius ] d π π e +3 sin+ d b Consider the region bounded by y + π, y arctan, and. Rotate this region about the vertical line 6. Set-up, BUT DO NOT EVALUATE!!, the integral to compute the volume of the resulting solid using the Cylindrical Shells Method. Sketch the solid, along with one of the approimating shells. See me for a sketch. V π radius height d π 6 + π arctan d c Consider the region bounded by y ln, y, and e. Rotate this region about the y-ais. COMPUTE the volume of the resulting solid using the Cylindrical Shells Method. Sketch the solid, along with one of the approimating shells. See me for a sketch. V e π radius height d π e ln d π e ln d

5 [ π π [e e π [e e ] e ln e lne+ + e π [e e + e 5 ] π [ ln+ + ] [ 3e π 5 ] π ] e ln+ ] [ 3e ] 5 ********************************************************** ** I.B.P. ln d ln d ln u ln dv d du d v 7. [ Points] Consider the Parametric Curve represented by 3 t and y e t +e t. a Compute the arclength of this parametric curve for t. d dy L + dt dt dt +e t e t dt +e t +e t dt e t ++e t dt e t +e t dt e t +e t dt e t e t e e e e e e e e b Set-up, BUT DO NOT EVALUATE!!, the definite integral representing the surface area obtained by rotating this curve about the -ais, for t. d dy S.A. πyt + dt πyt dt dt +e t e t dt π e t +e t e t +e t dt π e t +e t e t +e t dt from partb above. *************************************************************************** OPTIONAL BONUS Do not attempt this unless you are completely done with the rest of the eam. *************************************************************************** 5

6 OPTIONAL BONUS # Compute the sum n 3 First n comes from n 3 n with Recognize n 3 n n 3 n n n n d n n d d n n n d n n n d d n n d d d d d d n n d d n n d d d d d d d n d d d d d d d d d d d d d d d d d d d d + d d + d 3 d d Finally, substituting, we get n 3 n n 3 n