Elementary Textbook On the Calculus
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n^+l 138 DIFFERENTIAL CALCULUS 75. Interval of convergence. If the terms u^, u^, ••• of a given series are functions of a variable x, then the series will usually converge for . Some values of x and diverge for all others. In such a case the problem is to determine the interval of conver- gence, that is, the range of values of x for which the series is convergent The following examples will illustrate the method of procedure.
Ex. 1. Determine the interval of convergence of the series 1 + X + 2 ...x2 + 3 x3 + ... + rix" + ••• . In this case m„ = (^ — l)a;""~^ and m„+i = nx"..
Hence, ""^ = -J^ ^ ^^x.
w„ (n-l)x''-i n-l According to the ratio condition for convergence, it is necessary that this ratio shall be numerically less than 1 for all values of n exceeding a fixed number k. As r^ increases, the fraction n — 1 approaches unity. Hence if | x | has any fixed value less than 1, the given series is absolutely convergent. The interval of convergence is defined by the inequalities — 1
What to read after Elementary Textbook On the Calculus?
You can find similar books in the "Read Also" column, or choose other free books by Virgil Snyder to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
n^+l 138 DIFFERENTIAL CALCULUS 75. Interval of convergence. If the terms u^, u^, ••• of a given series are functions of a variable x, then the series will usually converge for . Some values of x and diverge for all others. In such a case the problem is to determine the interval of conver- gence, that is, the range of values of x for which the series is convergent The following examples will illustrate the method of procedure.
Ex. 1. Determine the interval of convergence of the series 1 + X + 2 ...x2 + 3 x3 + ... + rix" + ••• . In this case m„ = (^ — l)a;""~^ and m„+i = nx"..
Hence, ""^ = -J^ ^ ^^x.
w„ (n-l)x''-i n-l According to the ratio condition for convergence, it is necessary that this ratio shall be numerically less than 1 for all values of n exceeding a fixed number k. As r^ increases, the fraction n — 1 approaches unity. Hence if | x | has any fixed value less than 1, the given series is absolutely convergent. The interval of convergence is defined by the inequalities — 1
What to read after Elementary Textbook On the Calculus?
You can find similar books in the "Read Also" column, or choose other free books by Virgil Snyder to read onlineMoreLess
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