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The Ratio Calculator quickly finds equivalent forms of ratios and performs various operations, including simplifying, scaling up, and scaling down, with detailed calculations shown.
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Use our Radius of Convergence Calculator to find the radius within which a power series converges. The calculator also provides step-by-step solutions to help you understand the process and verify results.
"The radius of convergence is the maximum distance from the center of a power series within which the series converges."
For a series centered at x = a, the radius of convergence R determines the behavior of the series:
The two most common methods are the ratio test and the root test.
This test is useful when consecutive terms of the series are easily compared:
L = limn→∞ |an+1 / an|
The series converges if L < 1, and diverges if L > 1.
Use the root test when terms involve powers of n:
L = limn→∞ |an|1/n
Again, the series converges if L < 1.
Find the radius of convergence of the series:
∑n=1∞ (x - 3)n / n
Let Cn = (x - 3)n / n. Using the ratio test:
L = limn→∞ |Cn+1 / Cn| = limn→∞ |((x-3)n+1 / (n+1)) × (n / (x-3)n)| = limn→∞ |x-3| × (n / (n+1)) = |x-3|
Convergence requires |x - 3| < 1. Therefore:
R = 1
The interval of convergence is:
2 < x < 4
Yes. If the series converges at only one point, the radius of convergence is zero. For any other value of x, the series diverges.
If the limit in the ratio or root test approaches zero, the radius of convergence is infinite, meaning the series converges for all x.
Yes. Some series, like exponential series, converge for all complex numbers, resulting in an infinite radius of convergence.
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