* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Functional decomposition wikipedia, lookup

Abuse of notation wikipedia, lookup

Approximations of π wikipedia, lookup

Karhunen–Loève theorem wikipedia, lookup

Elementary mathematics wikipedia, lookup

Central limit theorem wikipedia, lookup

Collatz conjecture wikipedia, lookup

Proofs of Fermat's little theorem wikipedia, lookup

Large numbers wikipedia, lookup

Uniform convergence wikipedia, lookup

Non-standard calculus wikipedia, lookup

Law of large numbers wikipedia, lookup

Transcript

Worksheet 12 MATH 3283W Fall 2012 1. Show that the sequence an = sin(sin(. . . sin(1))) converges and find its limit. {z } | n Hint. Recall that sin θ < θ for any angle θ > 0. Definition. Given a sequence of real numbers (an ), we define a new sequence of partial n P ak . If this new sequence (sn ) converges, we say that the series sums (sn ) by the rule sn = k=1 ∞ P ak converges. In this case the limit lim sn is called the sum of the series. Otherwise, the k=1 series is said to diverge. Example1 . Let an = rn for some fixed real number r and n ∈ N ∪ {0}. Then the series ∞ P k r is called the geometric series and its n-th partial sum is equal to sn = 1 + r + · · · + rn = k=0 1−rn+1 . 1−r If |r| < 1, then this series converges and its sum is equal to 2. Find the formula for the partial sums of the series converges. 3. Find the formula for the partial sums of the series ∞ P k=2 ∞ P k=1 series converges. 1 Example 32.7 1 5k 1 1−r + 1 7k 1 (k+1)(k+3) and show that it and show that the ∞ P 1 4. Find the formula for the partial sums of the series and show that the series k2 +3k+2 k=1 converges. 5. Let ∞ P ak = a1 + a2 + · · · + ak + . . . be a convergent series with ak ≥ 0 for all k ∈ N. k=1 Prove that the series ∞ P a2k = a2 + a4 + · · · + a2k + . . . converges as well. Will it be k=1 true without the assumption that ak ≥ 0? 2