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By Harold Jeffreys

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The sequence is called finite or infinite according as there are or are not a finite number of terms. The sequence u1, u2, u3, . . is is also designated briefly by {un}. EXAMPLES. 1. 2. The set of numbers 2, 7, 12, 17, . , 32 is a finite sequence; the nth term is given by un = 2 + 5 (n – 1) = 5n – 3, n = 1, 2, . , 7. The set of numbers 1, 1/3, 1/5, 1/7, . . is an infinite sequence with nth term un = 1/(2n – 1), n = 1, 2, 3, . . Unless otherwise specified, we shall consider infinite sequences only.

If for any positive number ⑀ we can find a positive number N depending on ⑀ such that ⏐un – l⏐ < ⑀ for all integers n > N. In such case we write lim n→∞ un = l. EXAMPLE. If un = 3 + 1/n = (3n + 1)/n, the sequence is 4, 7/2, 10/3, . . and we can show that lim un = 3. n→∞ If the limit of a sequence exists, the sequence is called convergent; otherwise, it is called divergent. , if a limit exists, it is unique. 8. A more intuitive but unrigorous way of expressing this concept of limit is to say that a sequence u1, u2, u3, .

1)n/(2n – 1), . . (c) 1, –3, 5, –7, . . , (–1)n–1 (2n – 1), . . 3 5 7 2 3 4 5 (b) , – , , – , . . , (–1)n+ 1 (n + 1)/(n + 2), . . (d) 1, 4, 1, 16, 1, 36, . , n1 +(–1)n, . . 3 4 5 6 1 Ans. 51. Prove that a bounded sequence {un} is convergent if and only if lim un = limun. 52. Find the sum of the series ∑ n =1 Ans. 2 ⎛2⎞ ⎜ ⎟ ⎝3⎠ n . 53. Evaluate ∑ (−1) n −1 / 5n . n =1 Ans. =∑ = − . = 1. 54. 55. Prove that multiplication of each term of an infinite series by a constant (not zero) does not affect the convergence or divergence.

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Asymptotic approximations by Harold Jeffreys


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