Find the radius, then test each endpoint

Try

The geometric series Σxⁿ converges for |x|<1 but not at either endpoint. Now examine Σxⁿ/n. The extra denominator changes the endpoint behavior even though the radius remains 1. Predict why a radius alone cannot describe every input where a series converges.

Predict

For Σ(x−2)ⁿ/(n3ⁿ), identify the center before taking ratios. Predict the interval where the geometric part has magnitude below 1. Keep the candidate endpoints on a separate list; they are questions to test, not automatic inclusions or exclusions.

Experiment

The absolute ratio tends to |x−2|/3 for fixed x. Its value is below 1 when −1<x<5. Substitute the endpoints into the original terms.

Input	Resulting series, n starting at 1	Behavior
−1	Σ(−1)ⁿ/n	conditionally convergent
2	every term zero	convergent
5	Σ1/n	divergent

The full interval is [−1,5), while its center is 2 and radius 3. Draw a number line with an included dot at −1 and excluded dot at 5. The asymmetric endpoint symbols are mathematically significant, not presentation details.

Understand

A power series centered at a has the form Σcₙ(x−a)ⁿ. There is a radius R, possibly zero or infinity, such that it converges absolutely for |x−a|<R and diverges for |x−a|>R. At a finite positive boundary radius, either endpoint may converge or diverge independently. The interval of convergence includes precisely the endpoints that pass separate tests.

Treat x as fixed when taking n to infinity. For Σxⁿ/n!, the ratio is |x|/(n+1)→0 for every finite real x, so R is infinite. This does not assert uniform accuracy of a particular finite polynomial over every real input. For Σn!xⁿ, every nonzero fixed x makes the ratio grow without bound, while x=0 leaves only the constant term. Its radius is zero.

If the ratio test yields |x−a|/R, equality makes its limit 1, where the test is inconclusive. Return to the original series rather than manipulating the inconclusive inequality. At one endpoint signs may alternate; at the other a harmonic or p-series may appear. Indexing and constant factors still matter for values, although finite nonzero factors do not change convergence classification.

A translated center changes which inputs correspond to the same geometric distance. In Σ(x+4)ⁿ/(n!2ⁿ), the center is −4, not 4. The factorial still supplies infinite radius. Write x+4 as x−(−4) to make the structure explicit. Comparing series solely by the visible sign inside parentheses is a common source of reversed interval endpoints.

The convergence interval describes an infinite object. A graph of a ten-term polynomial exists at every real x, even where the infinite series diverges. Its finite values outside the interval are not evidence that the series converges there. A numerical display may remain modest for a while before a growing tail becomes visible.

Apply

For each practice series, state center, open radius condition and separate endpoint conclusions. Show how the absolute ratio is simplified. In the radius-zero example, inspect the center directly rather than applying a formula that divided by x. In the infinite-radius example, keep the quantifier “for each fixed x” visible.

Continue

Next choose power-series coefficients so that successive derivatives match a function at a center. Matching derivatives constructs Taylor polynomials; remainder control determines whether their infinite series actually represents the function.