Question

Can someone show me how to do question 2a and all 3 and 4?

I tried ratio test for 2a, but if x = 0, rhe proof doesn't work.

Thanks a lot.

2. Prove the following. (a) The series o converges for all 3 € R. (b) For n e N and k € {2,..., n}, the binomial coefficient (7) satisfies *)-(-5) (-)-(---) (c) For x > 0, the sequence (1 + 5)" is monotone increasing and bounded above by to (Hint. Write down the binomial expansions of (1+A)" and (1 + 11 )"+1, and for 0 <k <n compare the coefficients of xk in the two expansions.) In particular, the sequence (1 + )" converges; the number e is defined to be its limit. 3. (Optional) Let e = limn+ (1+ )" (a) For x > 0, prove that limn+ (1+ )" = e". (Hint. First consider rational x > 0.) (b) Deduce that limn+(1+)" = e* for all x € R. (Hint. For x > 0, consider the sequence (1+,-)" and its reciprocal.) 4. Let f : (a,b) + R be an increasing function and put D = f(a,b)). (a) Use the theorem proved in the lectures (R] (4.29)) and the Intermediate Value Theorem to show that D is an interval if and only if f is continuous. (b) Suppose f is strictly increasing and continuous. Prove that f-1: D+ (a,b) is strictly increasing and continuous

Answer 1

Given series is ** X HER SW! uy - X ht! 11 Unti (n+i) ni ein (+1) lin > Alip The given series converges by ratio test * skacial cases When y=0 Colt =of being - LO KI KT being constant, converge it co ty CSScanned with CamScanner

Given series is ** X HER SW! uy - X ht! 11 Unti (n+i) ni ein (+1) lin > Alip The given series converges by ratio test * skacial cases When y=0 Colt =of being - LO KI KT being constant, converge it co ty CSScanned with CamScanner

4 Solne Let I=(a, b) Then, fi IR is increasing fun D = f(5). where some @. By contradiction! Ang ver is not an intervalles J a, b, c ET with a ,cET & b¢ T. of f(a) is not an interval, find a, y EJ with (n=acc ce=hy), Beca,c) \ f (I), But this contradicts the Intermediate value Property , because ove every be ( f(x), f(y)) is equal to & (Z) for some z between a y, which makes ZE I & hence 6= f(z) € f (I)., HP & Scanned with CamScanner

-3. (a) lim n = el La let (17) nyo ( 14 (do show) A = lim dake in bothside ho » Lad = lim nln (172) Nyoo =lim in (174) for) Yn By L' Hospidal wule x) In A lim {1+Y/n] (-102) Alim (14an) {1+0) 2. Die limo (17) 1=0x FREIR n2 nto lietot n too x =) xto CS Scanned with CamScanner
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