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Worksheet 9.1—Sequences & Series: Convergence & Divergence Show all work. No calculator except unless specifically stated. Short Answer 1. Determine if the sequence 2 lnn n ½ ®¾ ¯¿ converges. 2. Find the nth term (rule of sequence) of each sequence, and use it to determine whether or not the sequence converges. (a) 2, 3 4, 4 9, 5 16, 6 ...
Give the precise definition of a sequence. What does it mean to say that lim f(x) = L when a = 1? Does this di↵er from. x!a. lim f(n) = L? Why or why not? n!1. (c) What does it means for a sequence to converge? Explain your idea, not just the definition in the book. (d) Sequences can diverge in di↵erent ways.
You may be asked to recall and apply de nitions, nd limits of sequences (through calculations that may involve several steps), nd formulas to represent terms of a sequence, and/or apply theorems to determine convergence or divergence.
Determine whether a sequence converges or diverges, and if it converges, to what value.
Math 115 Exam #1 Practice Problems. For each of the following, say whether it converges or diverges and explain why. 1. P∞ n3 n=1 n5+3. Answer: Notice that. n3 n3 1. n5 < = + 3 n5 n2. for all n. Therefore, since P 1 n2 converges (it’s a p-series with p = 2 > 1), the series P n3 also n5+3 converges by the comparison test.
Let f : X ! Y be a function of topological spaces. Then f is called sequentially continuous if for any sequence (xn)n2N in X and any limit x1 of the sequence, the sequence (f(xn))n2N in Y converges to the point f(x1).
Use the indicated test for convergence to determine if the series converges or diverges. If possible, state the value to which it converges. (a) Geometric Series: ∞. (c) p-series: ∑ n − 2/3. n = 1 ∞ en. (e) Direct Comparison: ∑. n = 1 n.
