How to Calculate Sums of Infinite Series
An infinite series adds an infinite number of terms together. The terms follow a pattern given by the series. The sum of the terms in the series may converge to a distinct value, or may diverge to infinity. There are several mathematical tests to determine whether a given infinite series converges or diverges, and several known formulas for solving the sums of particular forms of infinite series.
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Instructions
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Test for series convergence. If a series diverges, no sum can be calculated. There are several tests for convergence, depending on the form of the infinite series:
Divergence Test: Assume a series summing up terms in the form A1, A2, A3 ... An for any value of n. If the limit of An as n goes to infinity ≠ 0, then the series diverges.
Integral Test: Assuming a series summing up terms in the form f(n), then the series converges only if the integral of f(x) from 1 to infinity also converges.
Comparison Test: Assume two series, one with terms of the form An, the other with terms of the form Bn, and assume that for every value of n, An > Bn. If the An series converges, so does the Bn series, but if the Bn series diverges, so does the An series.
Limit Test: Assuming two series, one with terms of the form An, the other with terms of the form Bn, then if the limit as n goes to infinity of An/Bn = L, with 0<L<infinity, then both the An and Bn series either diverge or converge (and if you know one converges, you know the other one does as well).
Ratio Test: Assume a series of terms in the form An, and that the limit as n goes to infinity of An+1/An = L. If L < 1, the series converges; if L> 1, the series diverges; if L = 1, the test is inconclusive.
Alternating Series Test: Assuming a series of terms in the form (-1)^n * An, where An is always positive, then the series converges if An+1 < An and the limit of An as n goes to infinity = 0.
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Identify the type of converging series. Some common forms of converging infinite series have known sums. If you can match the series in question to one of these forms, calculating the sum should be easy.
Geometric Series: This series has terms in the form An = C*r^n (where C and r are constants). Based on the above tests, this series converges if r<1. If it converges, its sum is C/(1-r).
Harmonic Series: This series has terms in the form An = 1/n. Based on several of the above tests, this series diverges, and its sum is infinite.
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Solve the formula to calculate the sum. Using the solution forms in the previous step, plug in the particular constants from the series in question to solve for the sum.
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Tips & Warnings
Even if the series in question is not presently in one of the forms for which solutions are known, you may be able to manipulate the form to fit it into one of these boxes using factoring or other techniques.
The convergence tests and examples of solution forms are not exclusive, but merely an example of the most common types of infinite series and their sums.
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