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Which is the function of a power series?
Multiplying both sides by 1 / 3 gives ∞ ∑ n = 0(x + 2)n 3n + 1 = 1 1 − x, the same function as before. For what values of x does this series converge?
Which is an example of convergence of a power series?
Before getting into some examples let’s take a quick look at the convergence of a power series for the case of x = a x = a. In this case the power series becomes, and so the power series converges. Note that we had to strip out the first term since it was the only non-zero term in the series.
When does a power series converge in calculus?
We’ll have L = ∞ > 1 provided x ≠ − 1 2. So, this power series will only converge if x = − 1 2. If you think about it we actually already knew that however. From our initial discussion we know that every power series will converge for x = a and in this case a = − 1 2.
When does a power series converge in the ratio test?
By the ratio test, the power series converges if 0 ≤ rR, which proves the result. The root test gives an expression for the radius of convergence of a general power series.
Last Update: Oct 2021
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