1987 | OriginalPaper | Buchkapitel
The Fascination of Infinite Series
verfasst von : Eli Maor
Erschienen in: To Infinity and Beyond
Verlag: Birkhäuser Boston
Enthalten in: Professional Book Archive
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
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 aninfinite 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.”