Find the sum of the infinite series if it exists.
There are different kinds of series. There's harmonic, arithmetic, geometric.
Infinite arithmetic series will never have sums. This is because if the common difference is non-zero (infinite series), there's always a next number, and the series will not converge.
In the case of a geometric series, infinite kinds of these series will have sums as long as the series will converge. The series will converge if the absolute value of the common ratio is less than 1. In such case, the sum of the infinite geometric series is given by the following:
`S_infty = (a_1)/(1-r)` where `a_1` is the first term, and `|r| < 1` .
In the case of a harmonic series, it is more complicated. There are harmonic series that converge, and there are those that diverge, and techniques in calculus are used to determine convergence or divergence. An example of a harmonic series (infinite) that has a sum is the alternating series:
`sum^infty_(k=1) ((-1)^(n+1))/n` whose sum is equal to `ln(2)` .
An infinite series is given by all the terms of an infinite sequence, added together. ∑_(k=1)^∞1/2k =1/2 + 1/4 + 1/8 +……….. =?
Let us try adding up the first few terms and see what happens. If we add up the first two terms we get
1/2 + 1/4 = 3/4
The sum of the first three terms is 1/2 + 1/4 + 1/8 = 7/8
These sums of the first terms of the series are called partial sums. The first partial sum is just the first term on its own, so in this case it would be 1/2. The second partial sum is the sum of the first two terms, giving 3/4. The third partial sum is the sum of the first three terms, giving7/8, and so on.