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If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge. Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:
If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge. The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. [1]
The geometric series on the real line. In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as
[4] [10] This is because if Σa n is convergent, a second convergent series Σb n can be found which converges more slowly: i.e., it has the property that lim n->∞ (b n /a n) = ∞. Furthermore, if Σa n is divergent, a second divergent series Σb n can be found which diverges more slowly: i.e., it has the property that lim n->∞ (b n /a n) = 0.
If such a limit exists and is finite, the sequence is called convergent. [2] A sequence that does not converge is said to be divergent. [3] The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. [1]
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
Such a series = is considered to be convergent if and only if the sequence of partial sums is convergent, where = =. It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers p > q , {\displaystyle p>q,} s p − s q = ∑ n = q + 1 p x n . {\displaystyle s_{p}-s_{q}=\sum _{n=q+1}^{p}x ...
The addition of two divergent series may yield a convergent series: for instance, the addition of a divergent series with a series of its terms times will yield a series of all zeros that converges to zero. However, for any two series where one converges and the other diverges, the result of their addition diverges.