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Power Series Definition

Theory

A power series is an infinite series of the form

∑n=0∞cn(x−a)n,\sum_{n=0}^{\infty} c_n (x-a)^n,

where aa is called the center of the series and cnc_n are constants called the coefficients.
If a=0a=0, the series is called a power series in xx.

Since xx is a variable, a power series may converge for some values of xx and diverge for others. A power series always converges at its center x=ax=a.

Exercises

Exercise 1
Question

Identify the center of the power series

∑n=0∞(x+3)n(−1)nn!.\sum_{n=0}^{\infty} \frac{(x+3)^n(-1)^n}{n!}.
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Solution

Rewrite (x+3)n=(x−(−3))n(x+3)^n=(x-(-3))^n.
Thus, the center is

a=−3.a=-3.
Exercise 2
Question

Consider the power series

∑n=1∞(x−5)nn2.\sum_{n=1}^{\infty} \frac{(x-5)^n}{n^2}.

Determine the convergence at x=6x=6 and x=3x=3.

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Solution

At x=6x=6:

∑n=1∞(6−5)nn2=∑n=1∞1n2,\sum_{n=1}^{\infty} \frac{(6-5)^n}{n^2} =\sum_{n=1}^{\infty} \frac{1}{n^2},

which is a convergent pp-series with p=2p=2.

At x=3x=3:

∑n=1∞(3−5)nn2=∑n=1∞(−2)nn2.\sum_{n=1}^{\infty} \frac{(3-5)^n}{n^2} =\sum_{n=1}^{\infty} \frac{(-2)^n}{n^2}.

Consider absolute convergence:

∑n=1∞∣(−2)nn2∣=∑n=1∞2nn2,\sum_{n=1}^{\infty} \left|\frac{(-2)^n}{n^2}\right| =\sum_{n=1}^{\infty} \frac{2^n}{n^2},

which diverges since exponential growth dominates polynomial decay.

However, since

∑n=1∞(−2)nn2\sum_{n=1}^{\infty} \frac{(-2)^n}{n^2}

is an alternating series with terms not approaching zero fast enough to offset exponential growth, the series diverges.

Conclusion:
Converges at x=6x=6, diverges at x=3x=3.