91Ó°ÊÓ

When dealing with functions expressed as a series, like power series, integrating the series directly can simplify the process. This technique involves integrating each term in the series independently, ensuring that the operation can be performed in a manageable manner.
The integral of a series is important in calculus as it allows us to find the antiderivative or area under the curve represented by the series. For instance, if we have a series representation of a function, say \( f(t) = \sum_{n=0}^{\infty} a_n t^n \), we can integrate it from 0 to \( x \) as:\[ F(x) = \int_{0}^{x} f(t) \, dt = \int_{0}^{x} \left( \sum_{n=0}^{\infty} a_n t^n \right) \, dt \]
The integration is performed term by term, leading to an integrated series:
- Convert each term to its antiderivative.
- Evaluate the limits of integration for each term.

This procedure transforms a complex integral into a sequence of simpler, more manageable calculations.

The inverse hyperbolic tangent, denoted as \( \tanh^{-1} x \), is a function related to hyperbolic trigonometric functions. It plays a critical role in some mathematical analyses due to its unique properties.
One of the defining characteristics of \( \tanh^{-1} x \) is its relationship to the logarithmic expression:\[ \tanh^{-1} x = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right) \]
However, it is often more convenient to express \( \tanh^{-1} x \) in terms of a Maclaurin series for easier computation over specific intervals, particularly when \( |x| < 1 \). This series can be expressed as:
- \( x + \frac{x^3}{3} + \frac{x^5}{5} + \frac{x^7}{7} + \cdots \)
- Which is \( \sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1} \)
This provides a polynomial approximation that becomes increasingly accurate as the number of terms increases.

Term-by-term integration is a process where each component of a power series is integrated independently. This is a valid approach under certain conditions of convergence. When each term of a series is an integral itself, it allows for:
- Easier computational steps, focusing on each power individually.
- Clear insights into the convergence of the resulting series.
As seen in the exercise, integrating \( \sum_{n=0}^\infty t^{2n} \) term by term resembles basic polynomial integration. Each term \( t^{2n} \) becomes \( \frac{t^{2n+1}}{2n+1} \), leading to:
\[ \int t^{2n} \, dt = \frac{t^{2n+1}}{2n+1} \]
For the function \( F(x) \), integrating yields a new series in \( x \):
\[ F(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1} \]
This highlights the seamless integration enabled by term-wise calculation.

A power series is a way of expressing a function as an infinite polynomial, providing a powerful tool for approximating functions and performing calculus operations. In its general form, a power series is written as:
\[ f(t) = \sum_{n=0}^{\infty} a_n t^n \] where \( a_n \) are coefficients that determine the function's specific behavior.
Power series expansions are particularly useful for:

• Representing complex functions as simpler polynomials.
• Facilitating easier computations than might be available through the original function.
• Smoothing computations of derivatives and integrals.

A Maclaurin series is a special kind of power series centered at zero,
offering a direct representation of a function's behavior near that point.
In the example given, expressing \( \frac{1}{1-t^2} \) as \( \sum_{n=0}^{\infty} t^{2n} \) helps simplify integration processes as each term is easily manageable.