Suppose an- is a decreasing sequence of non-negative numbers (that is, 0 S an+1 S an...
Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise 21: by A =ļa, and An = zou-a + ax=a + zam for k21. Prove that an converges to some S if and only if Ax converges to S. N=0 k=0 Exercise 22: (Cauchy condensation test) Let (an) be a sequence such that 0 < antı san a) Show n=0 n=1 Hint: Recall the proof of convergence of for p > 1. Ren for...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
2. Suppose that (an), İs a sequence of complex numbers such that there exists a positive number 0 such that for all NEN an M (i) Show that (ON)N converges to a number . (ii) Show that sx -2Nan for N E N is a Cauchy sequence 2. Suppose that (an), İs a sequence of complex numbers such that there exists a positive number 0 such that for all NEN an M (i) Show that (ON)N converges to a number...
+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...
please answer questions with details and clear steps and clear hand writing (8,9,10,11 S. Find the values of p for which the series is convergent. /2 9. Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? a s " 2n +3 10. Test the series for convergence or divergence. 72 11. a) Use the integral test to show that the series converges In+1 b) Find s,o c) Use the error bounds from the integral...
1. Let {n} be a sequence of non negative real numbers, and suppose that limnan = 0 and 11 + x2 + ... + In <oo. lim sup - n-00 Prove that the sequence x + x + ... + converges and determine its limit. Hint: Start by trying to determine lim supno Yn. What can you say about lim infn- Yn? 3 ) for all n Expanded Hint: First, show that given any e > 0 we have (...
Question 1 3+cos(n) 2n X Which of the following properties hold for the sequence an for n 2 1? l. Bounded Il. Monotonic IIl. Convergent Selected Answer a. I only a. I only b. Il only c. I and Il only d. I and Ill only e. I, II, and III Remember what these conditions mean: Bounded means all terms of the sequence have to lie within a specific range of values. Monotonic means the sequence is ALWAYS increasing or...
41. Let (an) be a sequence of strictly positive real numbers and Sn = ak (a) Suppose that the series Σ an/ S,. an is convergent, determine the nature of the series an is divergent, show that 00 (b) Suppose that the series 1 1 Sn-1 Sp an a/S Then deduce the nature of the series 41. Let (an) be a sequence of strictly positive real numbers and Sn = ak (a) Suppose that the series Σ an/ S,. an...
(1 point) Each of the following statements is an attempt to show that a given series is convergent or divergent using the Comparison Test (NOT the Limit Comparison Test) For each statement, enter C (for "correct") if the argument is valid, or enter I (for "incorrect") if any part of the argument is flawed. (Note: if the conclusion is true but the argument that led to it was wrong, you must enter I.) TL 1. For all n > 1....
Let Pbé à pósitive, continuous, and decreasing function for x 2 1, such that an-n). If the series an n 1 converges to S, then the remainder RN -S-Sw is bounded by Use the result above to approximate the sum of the convergent series using the indicated number of terms. (Round your answers to four decimal place Σ ,T, ten terms n2 +1' Include an estimate of the maximum error for your approximation. (Give your answer to four decimal places.)...