radius of convergence power series
A power series adds powers of x without end. It is written 1 + x + x² + x³ + .... An infinite sum can still be a finite number, and the reason is distance. Inside a radius R from the center the terms shrink and the sum settles on one value. Step outside R and the terms grow, so the sum swells without end. That boundary distance R is the radius of convergence. The cleanest example is the geometric series 1 + x + x² + ... = 1/(1-x), and here R=1. The boundary points x=±R have to be checked on their own. The pictures below show the partial sums hugging the limit inside and swinging outside, how the ratio test pins down R, and why the boundary is delicate. The gold band in the later picture is what separates the inside from the outside.
Gold is the partial sum SN(x) = 1 + x + ... + xN, and gray is the true limit 1/(1-x). Raise the number of terms N. On the interval (-1, 1), as N grows the gold curve settles onto the gray one. On the side |x|>1 the opposite happens. As you raise N the gold curve swings more steeply up and down and leaves the gray curve. You can also move the probe x. Inside the band, S converges to 1/(1-x). Outside, the readout says the series diverges. Whether the series lives or dies with distance is the whole of this picture.
The radius comes from the ratio test. Look at the size ratio of two successive terms, the next term divided by the current one. For Σ xⁿ that ratio is constantly |x|. The bars are the term sizes |x|ⁿ. Move the x slider and the difference shows. When |x|<1 the ratio is below 1, the bars shrink in turn, and the color is green for convergence. When |x|>1 the bars grow and the color is red for divergence. The ratio equal to exactly 1, which is |x|=1, is the boundary between convergence and divergence, the radius itself. On that boundary the bars are gold, and the ratio test says nothing more. What happens there continues in the next two pictures.
Now set the radius directly on the number line. For the series Σ(x/a)ⁿ the radius of convergence is R=a. The gold band around the center 0 is the interval of convergence (-R, R). Raise R with the slider and the band widens. Lower it and the band narrows. Move the probe x. Inside the band the point is green (|x|<R, converges). Outside the band it turns red and diverges. Convergence is the one-line condition that the distance |x| is closer than the radius R. Only the absolute value counts, so the region is always an interval symmetric about the center. A point sitting on the edge is the boundary, neither inside nor outside.
Equal radii can still have different fates on the boundary. You can choose among three series. All three have radius 1, yet they behave differently at the boundary x=±1. Σxⁿ diverges at both ends. Σxⁿ/n converges at x=-1, where the terms alternate, but at x=+1 it becomes the harmonic series and diverges. Σxⁿ/n² shrinks faster, so it converges at both ends. The left panel and the right panel are those two ends, and the bars are the running partial sums. When they settle, the panel is green for convergence. When they drift or swing without end, it is red for divergence. The boundary is where the ratio test lets go, so each series has to be read on its own.
Last, return to the geometric series that anchors the rest. 1 + x + x² + ... = 1/(1-x), but only when |x|<1. Each bar is one term xⁿ. They stack from the top, and the sum gathers toward the green dashed line at 1/(1-x). Set the ratio x to a positive value such as 0.6 and the gold bars get shorter on the same side and settle. Make x negative and the bars stand on alternate sides of the baseline, the negative terms in blue, canceling each other, and still, because |x|<1, they settle on one value. The infinite sum is finite only when the terms shrink fast enough, which means the distance stays inside the radius. The green dashed line is where the sum belongs, and the series has become finite once the bars have stacked up to that line.