Properties of Infinite Series

To determine the values of x, which lead to convergent series, we can apply the ratio test (Boas 1983), which states that if the absolute value of the ratio of the (n + 1) term to nth term approaches a limit e as n ao, then the series itself converges when e 1 and diverges when s 1. The test fails if e = 1. In the case of the Power Series, Eq. 3.15, we see 

within the interval of convergence, the original Power Series can be treated like any other continuously differentiable function. Such series formed by differentiation or integration are thus guaranteed to be convergent. Consider a series we have come to know very well 

Consider the Binomial series discussed earlier in Seetion 2J.2 

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Mathematicians have discovered some very interesting facts in their investigations upon the properties of infinite series. Many of these results can be employed as tests for the convergency of any given series. I shall not give more than three tests to be used in this connection. 

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