Interval of Convergence for Taylor Series
Introduction
When working with Taylor or Maclaurin series, it is not enough to know the formula. We
must also determine the interval of convergence — the set of x values for which the series
converges to the function. This ensures that our polynomial approximation is valid and
matches the original function.
Ratio Test Review
The Ratio Test is the standard tool for finding the radius of convergence R:
an+1
lim =L
n→∞ an
- If L < 1, the series converges. - If L > 1, the series diverges. - If L = 1, the test is
inconclusive.
an (x − c)n , applying the ratio test gives:
P
For a power series
an+1
lim |x − c| < 1
n→∞ an
which defines the radius of convergence R.
Interval of Convergence
- The series converges absolutely for |x − c| < R. - At the endpoints x = c ± R, convergence
must be checked separately. - Thus the interval of convergence is either (c − R, c + R),
(c − R, c + R], [c − R, c + R), or [c − R, c + R] depending on endpoint behavior.
Detailed Examples
Example 1: ex
The Maclaurin series is:
1
Introduction
When working with Taylor or Maclaurin series, it is not enough to know the formula. We
must also determine the interval of convergence — the set of x values for which the series
converges to the function. This ensures that our polynomial approximation is valid and
matches the original function.
Ratio Test Review
The Ratio Test is the standard tool for finding the radius of convergence R:
an+1
lim =L
n→∞ an
- If L < 1, the series converges. - If L > 1, the series diverges. - If L = 1, the test is
inconclusive.
an (x − c)n , applying the ratio test gives:
P
For a power series
an+1
lim |x − c| < 1
n→∞ an
which defines the radius of convergence R.
Interval of Convergence
- The series converges absolutely for |x − c| < R. - At the endpoints x = c ± R, convergence
must be checked separately. - Thus the interval of convergence is either (c − R, c + R),
(c − R, c + R], [c − R, c + R), or [c − R, c + R] depending on endpoint behavior.
Detailed Examples
Example 1: ex
The Maclaurin series is:
1
