Question

The seriessec x = 1 + x²/2 + 5x⁴/24 + 61x⁶/720 + 277x⁸/8064 + ⋯converges to sec x for −π/2 \u003c x \u003c π/2.a. Find the first five terms of a power series for the function ln|sec x + tan x|. For what values of x should the series converge?

Accepted Answer

Recall the given power series for $$ \sec x $$:

$$
\sec x = 1 + \frac{x^{2}}{2} + \frac{5x^{4}}{24} + \frac{61x^{6}}{720} + \frac{277x^{8}}{8064} + \cdots
$$

and note that it converges for $$ -\frac{\pi}{2} \u003c x \u003c \frac{\pi}{2} . $$Use the identity:

$$
\ln|\sec x + 	an x| = \int \sec x \, dx
$$

This means the power series for $$ \ln|\sec x + 	an x| $$ can be found by integrating the power series for $$ \sec x $$ term-by-term. Integrate each term of the $$ \sec x $$ series separately:

- Integrate $$ 1  to $$get $$ x 
- $$Integrate $$ \frac{x^{2}}{2}  to $$get $$ \frac{x^{3}}{6} 
- $$Integrate $$ \frac{5x^{4}}{24}  to $$get $$ \frac{5x^{5}}{120} 
- $$Integrate $$ \frac{61x^{6}}{720}  to $$get $$ \frac{61x^{7}}{5040} 
- $$Integrate $$ \frac{277x^{8}}{8064}  to $$get $$ \frac{277x^{9}}{72576} $$ Write the resulting power series for $$ \ln|\sec x + 	an x|  as$$:

$$
\ln|\sec x + 	an x| = C + x + \frac{x^{3}}{6} + \frac{5x^{5}}{120} + \frac{61x^{7}}{5040} + \frac{277x^{9}}{72576} + \cdots
$$

where $$ C  is $$the constant of integration. Determine the interval of convergence: since the original $$ \sec x $$ series converges for $$ -\frac{\pi}{2} \u003c x \u003c \frac{\pi}{2} $$, the integrated series for $$ \ln|\sec x + 	an x| $$ will converge on the same interval $$ -\frac{\pi}{2} \u003c x \u003c \frac{\pi}{2} .$$

The series
sec x = 1 + x²/2 + 5x⁴/24 + 61x⁶/720 + 277x⁸/8064 + ⋯
converges to sec x for −π/2 < x < π/2.
a. Find the first five terms of a power series for the function ln|sec x + tan x|. For what values of x should the series converge?

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Key Concepts

Power Series Expansion

Relationship Between Functions and Their Derivatives

Interval of Convergence

The seriessec x = 1 + x²/2 + 5x⁴/24 + 61x⁶/720 + 277x⁸/8064 + ⋯converges to sec x for −π/2 < x < π/2.a. Find the first five terms of a power series for the function ln|sec x + tan x|. For what values of x should the series converge?