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The Fascination of Infinite Series

In: To Infinity and Beyond

Author

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  • Eli Maor

    (Oakland University, Department of Mathematical Sciences)

Abstract

A series is obtained from a sequence by adding up its terms one by one. From a finite sequence a1, a2, a3, …, a n we obtain the finite series, or sum, a1 + a2 + a3 + …, + a n . But for an infinite sequence a1, a2, a3, …, a n , …, a problem arises: How should we compute its sum? We cannot, of course, actually add up all its infinitely many terms; but we can, instead, sum up a finite, but ever increasing, number of terms: a1, a1 + a2, a1 + a2 + a3, and so on. In this way we obtain a new sequence, the sequence of partial sums of the original sequence. For example, from the sequence 1, 1/2, 1/3, …, 1/n, … we get the sequence of partial sums 1, 1 + 1/2 = 1.5, 1 + 1/2 + 1/3 = 1.83333 …, and so on. If this sequence of partial sums converges to a limit S, then we say that the infinite series a1 + a2 + a3 + … converges to the sum S. For the sake of brevity, we also use the phrase, “the series has the (infinite) sum S.”

Suggested Citation

  • Eli Maor, 1987. "The Fascination of Infinite Series," Springer Books, in: To Infinity and Beyond, chapter 4, pages 25-28, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-5394-5_4
    DOI: 10.1007/978-1-4612-5394-5_4
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