sharks said:It's the simplest way of finding the limit of its partial sums, by the telescoping series.
In that case, it would not be a telescoping series.trap101 said:ok, but what happens if the partial fraction that you end up with doesn't have a minus sign in it, then how would the telescoping occur?
A series is a sum of terms, while a partial fraction is a fraction with a polynomial in the numerator and denominator. In other words, a series is a sum of fractions, while a partial fraction is a single fraction.
Partial fractions are used to simplify and solve complex series problems. By breaking down a single fraction into multiple partial fractions, we can often find a simpler solution or determine the convergence of a series.
To find the partial fraction decomposition of a series, we first factor the denominator of the fraction into linear or quadratic terms. Then, we solve for the coefficients of the partial fractions by equating the original fraction to the partial fraction decomposition and solving for the unknown coefficients.
Yes, partial fractions can be used to find the sum of an infinite series. By finding the partial fraction decomposition of the series, we can often simplify the problem and use known formulas to find the sum of the series.
In calculus, partial fractions are used to help solve integrals involving rational functions. By breaking down a complex rational function into simpler partial fractions, we can more easily find the antiderivative and solve the integral.