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**Unformatted text preview: **MAT 137Y1Y, April/May 2008 Examinations Page 11 of 20
4. (2%) (3) Give the formal deﬁnition of the statement: lim f (x) = L. JC—MI (7%) (3) Suppose )lcim f (x) = lim g(x) = L. Prove, using the formal deﬁnition, that
—>a x->a
lim (f(x) +g(x)) = 2L. x—>a Examination continues on the next page. MAT 137Y1Y, April/May 2008 Examinations Page 12 of 20
———-——————————————_________
5. Determine whether the following series converge or diverge and justify your answer. 1 (5%) (i) [gm Examination continues on the next page. MAT 137Y1Y, April/May 2008 Examinations Page 13 of 20
WW no 3k+27
5 .o ——-————————.
( %) (u) [£65k3+3k2 +6k+1 Examination continues on the next page. MAT 137Y1Y, April/May 2008 Examinations Page 14 of 20 6. Recall the geometric series 2 )d’, which converges for |x| < 1.
n=0 (3%) (a) If f (x) = 1 :x’ ﬁnd a power series representation for g(x) = 1 :x2 the general term. ; be sure to include (4%) (b) Find a power series representation for h(x) = arctanx. What is the radius of conver—
gence of this series? ________________________________._.___.__.——————-
Examination continues on the next page. MAT 137Y1Y, April/May 2008 Examinations Page 15 of 20
(Question 6 continued) . °° (—1)"
5‘7 U th 1t f artb,ﬁdth t 1 ——————.
( o) (c) Sing eresu o p ( ) n eexac vaue ofngban+1x2n+2> ______________________________________.____——
Examination continues on the next page. MAT 137Y1Y, April/May 2008 Examinations Page 16 of 20
7. Let the sequence {an} be given by a1 = x/Z an+1 = V2 +an for all n 2 1. (4%) (a) By induction, show that {an} is increasing. (4%) (b) By induction, show that an < 3 for all n 2 l. ________________________________—
Examination continues on the next page. MAT 137Y1Y, April/May 2008 Examinations Page 17 of 20
(Question 7 continued) (4%) (c) Show that lim an exists and ﬁnd the limit. n—mo ____—__.___—____—_—_———————————-—-————————-—— Examination continues on the next page. MAT 137Y1Y, April/May 2008 Examinations Page 18 of 20 8. Suppose f (x) has n + 1 continuous derivatives on an open interval I that contains the point
0, and furthermore, f’(0) = f”(0) = ... = Irv—”(0) = 0, but f(")(0) > 0,
where n 2 2. (3%) (a) Use Taylor polynomials to show that if n is even, then f has a local minimum at x = 0. (2%) (b) Show that if n is odd, then f has an inﬂection point at x = 0. ___________________.________________.___..___——————
Examination continues on the next page. MAT 137Y1Y, April/May 2008 Examinations Page 19 of 20 End of examination.
(This page is available for scrap work. Do NOT tear out this page!) MAT 137Y1Y, April/May 2008 Examinations Page 20 of 20 End of examination.
(This page is available for scrap work. Do NOT tear out this page!) ...

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