Question # 4. (5 marks) (a) State precisely our theorem on the convergence of telescoping series....
Question # 4. (5 marks) (a) State precisely our theorem on the convergence of telescoping series. (6) Evaluate the series, 6 2 21 ..+ 6 +... An? + 8n +3 (c) Consider the statement: Suppose we have a sequence, {an), and we can find another sequence, {Dm}, so that an = bn - bm+2.11 an converges then lim on erists. 00 Is this statement true? Either prove it (one may wish to consider similar proofs from our lessons), or construct...
does not Use the definition of convergence to explain why the seque converge to zero. – Definition 2.2.3 (Convergence of a Sequence). A sequence (an) converges real number a if, for every positive number €, there exists an N EN such that whenever n > N it follows that an - al < €. Ju To indicate that (an) converges to a, we usually write either lim an = a or (an) → a. The notation lim +00an =a is...
6. We want to use the Integral Test to show that the positive series a converges. All of the following need to be done except one. Which is the one we don't need to do? (a) Find a function f(x) defined on [1,00) such that f(x) > 0, f(x) is decreasing, and f(n) = a, for all n. (b) Show that ſ f(z) dr converges. (e) Show that lim Ss6 f(x) dx exists. (d) Show that lim sexists. 7. Suppose...
I'm having difficulty how many terms need to be added in. Test the series for convergence or divergence. 00 Σ (-1)" n2n n = 1 Identify bn. 1 n2" Evaluate the following limit. lim bn n → 00 0 Since lim bn O and bn + 1 s bn for all n, the series is convergent n00 If the series is convergent, use the Alternating Series Estimation Theorem to determine how many terms we need to add in order to...
Please show work, thank you. 1) Find a power series and radius of convergence for X x + 10 lim 2) Suppose that [bn+1xn+1 bnxn converges for all || < 2. Use the ratio test to conclude that <1 n-00 bn. -xh n=0 n + 1 converges for |«/ < 2.
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
10. Read through the following "e-free" proof of the uniform convergence of power series. Does it depend on limn→oo lan|1/n or lim supn→oo lan! an)1/n? Explain. 1.3 Theorem. For a given power series Σ ak-a)" define the number R, 0 < R < oo, by n-0 lim sup |an| 1/n, then (a) if |z- a < R, the series converges absolutely (b) if lz-a > R, the terms of the series become unbounded and so the (c) if o<r <...
Question 20 1 pts Does the following process sufficiently support the conditionally (-1)" convergence of a -? If not, state your reason. n+25 n=0 By alternating series test, we verify that 1 lim bn lim - 0 n- noon+25 1 1 1 . bn+1 < bn = (n+1)+25 n+26 n+25 Therefore, Σ (-1)" n+25 converges conditionally. n=0 HTML Editore Β Ι Ο Α I 를 를 를 트
Question # 2. (2 marks) Show that the ratio test fails to apply to the series, 7-n+(-1)", but that the root test does apply. Use the root test to determine if the series converges or not. n=0 Question # 3. (3 marks) Consider the power series, f(x) = į an(x + 1)". Suppose we know that f(-4), as a series, diverges, while f(2) converges. Determine the radius of convergence of the power series for f'(x). Precisely name the results we...
Question 4 can have more than 1 answer 4) The Comparison Test a) is a consequence of the Monotone Convergence Theorem. b) applies only to positive series. c) shows that if o San <br and if į bn converges, then an converges. d) None of the above. n=1 n=1 5) The Divergence Test n=1 a) shows that if lim n = 0, then į an converges. b) applies only to positive series. c) can be used to show that Enti...