3. Problem 3. Compute the numerical value of the following series: 4. Problem 4. Prove that...
+00 bn are series with positive terms an and (a) Suppose that O bn is convergent. Prove that if "limo and that I1 n= 1 +00 then Σ an is also convergent. (b) Use part (a) to show that the series converges t In (n) +00 bn are series with positive terms an and (a) Suppose that O bn is convergent. Prove that if "limo and that I1 n= 1 +00 then Σ an is also convergent. (b) Use part...
3. Evaluate the following convergent series (i.e. give a numerical value)
Is the following series cos n convergent or divergent? Prove your result. 2 if Σ an with an > o is convergent, then is Σ a.. always convergent? Either prove it or give a counter example. 3 Is the following series convergent or divergent? if it is divergent, prove your result; if it is convergent, estimate the sum. 4 Is the following series 2n3 +2 nal convergent or divergent? Prove your result.
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...
Problem 3: Let (w).>o be a sequence such that bn is convergent. Let (an)nzo to be a sequence such that lant - and by for all ne N. Prove that (an)nzo is a convergent sequence. (Hint: We did something similar in class before.)
a,b,c and d (-1) 4. (3 points each) Consider the series n° +2n +3 (a) Prove that this series converges absolutely. (b) Show that this series satisfies all three conditions of the Alternating Series Test. HI11-2212, JL ILG-2020 Test #3 (c) What value of n guarantees that the partial sum 8, approximates the sum of this series to within an accuracy of 0.01? (d) Find the sum of the series with this accuracy (by finding the appropriate partial sum sn,...
Problem 3. (1 point) Consider the series 4+ (-4)" 8" Which of the following statements accurately describes the series? A. It converges to B. It converges to C C. It converges to 38 21 D. It converges to E. None of the above, the series diverges. Problem 4. (1 point) Which of the following series converges by the Alternating Series Test? 48 (-1) (-1)" Vn - 5 00 sin(n) B. 7n2 nal (-1)"m? + in 3n2 + 7 Do D....
Match each of the following with the correct statement. A. The series is absolutely convergent. C. The series converges, but is not absolutely convergent. D. The series diverges. (1 point) Match each of the following with the correct statement. A. The series is absolutely convergent. C. The series converges, but is not absolutely convergent. D. The series diverges. Vi 2. n32 -1)" 3. 4. 5. 2n sin(5n) n2 (1 point) Match each of the following with the correct statement. A....
4 4. Answer the following questions. Justify your answers. A. (Spts) Determine if the series 2.0 is convergent or divergent. B. (8pts) Determine if the series is convergent or divergent. C (4pts) If an is a convergent series with nonnegative terms, what can you say about the convergence of the series 2? Is it convergent? Divergent? Or may converge or diverge depending on ay. Explain. 22 In
(a) State the First Comparison Test and show that the following series con- verges: O0 1 + cos ((2n +1)!) (b) Determine whether the following series converges (c) State the Integral Test and sketch its proof (d) Prove or disprove: If a series Σ001 an converges then Σηι an converges absolutely. e) Answer the following two questions without proof: For which r E R is the geometric series 0O convergent? What is the limit of the series in case of...