Math Problem Statement

what about arc tan

Solution

If you're asking about the series expansion or convergence properties of arctangent (e.g., arctan(x)\arctan(x)), let me provide insights for both:


1. Series Expansion of arctan(x)\arctan(x):

The function arctan(x)\arctan(x) has a Taylor series expansion for x1|x| \leq 1 given by: arctan(x)=n=0(1)nx2n+12n+1.\arctan(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1}.

Properties of the Series:

  • The series converges for all x1|x| \leq 1 (including at the endpoints, 1x1-1 \leq x \leq 1).
  • For x>1|x| > 1, the series does not converge.

2. Convergence of arctan(an)\sum \arctan(a_n):

If you're asking about the convergence of a series like: n=1arctan(an),\sum_{n=1}^\infty \arctan(a_n), the convergence depends on the behavior of the sequence ana_n. Since arctan(x)\arctan(x) is bounded (it approaches ±π2\pm \frac{\pi}{2} as x±x \to \pm \infty), this type of series must be examined carefully.

Example: n=1arctan(1n)\sum_{n=1}^\infty \arctan\left(\frac{1}{n}\right):

  • For large nn, arctan(1n)1n\arctan\left(\frac{1}{n}\right) \sim \frac{1}{n} (using the small-angle approximation).
  • Thus, the series behaves like 1n\sum \frac{1}{n}, which diverges (harmonic series).

General Rule:

  • If an0a_n \to 0 and decreases fast enough (e.g., exponentially), the series may converge.
  • If ana_n behaves like 1n\frac{1}{n} or slower, the series diverges.

3. Arctangent Integral:

If you're asking about integrals involving arctan(x)\arctan(x), the formula is: arctan(x)dx=xarctan(x)12ln(1+x2)+C.\int \arctan(x) \, dx = x \arctan(x) - \frac{1}{2} \ln(1 + x^2) + C.

Would you like a worked example of a specific question involving arctangent?


Practice Questions:

  1. Expand arctan(x)\arctan(x) into its Taylor series up to the fifth term.
  2. Determine the convergence of n=1arctan(1n2)\sum_{n=1}^\infty \arctan\left(\frac{1}{n^2}\right).
  3. Compute 01arctan(x)dx\int_0^1 \arctan(x) \, dx.
  4. Prove that n=1(1)n12n+1\sum_{n=1}^\infty (-1)^n \frac{1}{2n+1} converges and find its sum.
  5. Analyze the convergence of (\sum_{n=1}^\infty \arct

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Series Expansion
Convergence
Integration

Formulas

Taylor series for arctan(x): arctan(x) = ∑ (-1)^n (x^(2n+1) / (2n+1)) for |x| ≤ 1
Integral of arctan(x): ∫ arctan(x) dx = x arctan(x) - (1/2) ln(1 + x^2) + C

Theorems

Convergence of Taylor Series for |x| ≤ 1
Behavior of harmonic series and its divergence

Suitable Grade Level

Undergraduate Mathematics or Advanced High School (Grades 11-12)