Therefore sum of the given series is equal to 1.
Q9 (Approximation of π) (a) Show that 1/1 + t2 = 1 − t2 + t4 − ... + (−1)n−1 t 2n−2 + (−1)n t2n /1 + t2 for all t ∈ R and n ∈ N. (b) Integrate both side in (a), show that tan−1 (x) = x − x3/3 + x5 /5 − ... + (−1)n−1x 2n−1/ 2n − 1 + Z x 0 (−1)n t2n /1 + t2 dt. (c) Show that tan−1 (x) − ( x...
Find the sum of the series 2n=1 ni
2. Consider the system 2n+4x2 + 8x3 + 12n = b2 (a) Reduce A b to Row Reduced Echelon Form Rx-c (b) Find the condition on bi, b2, bs for Ax b to have a solution (c) Find the nullspace of A as the span of special solutions (d) Find a particular solution when b- 3 6 9 and the general solution.
Calculate the sum 〉 n 2n+1 Calculate the sum 〉 n 2n+1
12. For what values of r does the series (2n)!r" 22n(n!) converge absolutely? converge conditionally? diverge? n=1
1. Find 7(5") 3" 5n+12n+1 sin(n!) lim (a) im (b) n 2018 пH0 п2 n oo _ Зп? + п + 4(3") nt T lim sin (d) (с) lim 1 - 2n 3п — 2п n oo n oo 2. Let (an), be a sequence such that lim an n oo L. = 1 (a) What is lm an+1? пH00
(4) Guess a formula for the sum (2n 1) (2n +1) 1.3 3.5 Prove your guess using induction (4) Guess a formula for the sum (2n 1) (2n +1) 1.3 3.5 Prove your guess using induction
Evaluate the following infinite series. so (-1)"+12n-1 2n=0 n! 2e-2 e-2 0 -2e-2 0-5e-2 O none of the above or below
Please answer question 34 only 33. Find the sum of the series -correct to four decimal places. n" & 2n is comvergent. 34. (a) Show that the series (b) Deduce that lim0 (2n)! 35. Prove that if the series Σ-a, is absolutely convergent, then the series s (n + 1 33. Find the sum of the series -correct to four decimal places. n" & 2n is comvergent. 34. (a) Show that the series (b) Deduce that lim0 (2n)! 35. Prove...
11. (6 points) Find the sum of the following series: (a) Σ 2n +1 3η n=0 ΟΙ (5) Σ n! ΠΟ