Convergent Series Math
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In this section we will discuss in greater detail the convergence and divergence of infinite series.
Convergent series math. In comparison test we compare our series with a series whose convergence is already known to us. It s denoted as an infinite sum whether convergent or divergent. Alternating the signs of the reciprocals of positive integers produces a convergent series alternating harmonic series. Conversely a series is divergent if the sequence of partial sums is divergent.
If sumu k and sumv k are convergent series then sum u k v k and sum u k v k are. A necessary but not sufficient condition for a series of real positive terms to be convergent is that. If the aforementioned limit fails to exist the very same series diverges. We will also give the divergence test for series in this section.
The partial sums in equation 2 are geometric sums and. Formally the infinite series sum n 1 infty a n is convergent if the sequence of partial sums s n sum k 1 na k 1 is convergent. A series is said to be convergent if it approaches some limit d angelo and west 2000 p. 1 1 1 2 1 3 1 4 1 5 1.
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