Expressing an integral as an infinite series

Thread starter DrummingAtom
Start date Apr 19, 2011
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Infinite Infinite series Integral Series

In summary, an infinite series is a mathematical expression consisting of an infinite sum of terms. An integral can be expressed as an infinite series by using power series expansion or other methods. Expressing an integral as an infinite series can be useful in solving complex integration problems and providing a more accurate approximation. Some common examples include Maclaurin and trigonometric series, but there are limitations to this method such as limited convergence and accuracy.

Apr 19, 2011

DrummingAtom

Homework Statement

[tex] \int_0^x \frac{1-cos(t)}{t} [/tex]

Homework Equations

The Attempt at a Solution

I'm lost completely. If I separate it and then try integrating it has 0 for the ln(x) which has to be wrong.

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Apr 19, 2011

rock.freak667

Homework Helper

One equation that will be useful to you is that

[tex]cosx = \Sigma_{n=0} ^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}[/tex]

So you can cancel out the '1' from the numerator.

Related to Expressing an integral as an infinite series

1. What is an infinite series?

An infinite series is a mathematical expression consisting of an infinite sum of terms. It is represented in the form of ∑ a_n, where "∑" is the summation symbol, "a_n" represents the terms of the series, and "n" represents the index or position of the term.

2. How can an integral be expressed as an infinite series?

An integral can be expressed as an infinite series by using the concept of power series expansion. This involves finding the Taylor series expansion of the integrand and integrating it term by term. It is also possible to use other methods such as the Method of Undetermined Coefficients or the Method of Partial Fractions to express an integral as an infinite series.

3. What is the significance of expressing an integral as an infinite series?

Expressing an integral as an infinite series can be useful in solving complex integration problems. It can also help in evaluating the integral for a wider range of values, as the convergence of an infinite series can be easier to analyze than that of an integral. Additionally, infinite series can often provide a more accurate approximation of the integral.

4. What are some common examples of expressing an integral as an infinite series?

Some common examples of expressing an integral as an infinite series include the Maclaurin series for trigonometric functions such as sin(x), cos(x), and e^x, as well as the geometric series and the binomial series. These series have various applications in physics, engineering, and other branches of science.

5. Are there any limitations to expressing an integral as an infinite series?

Yes, there are certain limitations to expressing an integral as an infinite series. One limitation is that not all integrals can be expressed as an infinite series. Additionally, the convergence of an infinite series may be limited to a certain range of values, and the accuracy of the series may decrease as the number of terms increases.