The sum of the series (1)/(1+1^2+1^4)+(2)/(1+2^2+2^4)+(3)/(1+3^2+3^4)+…to n terms is (A) (n(n^2+

step 1 Note that the formula for the n's term tn equals 1 over 1 plus n squared plus n to the 4th and w

The sum of the series...

The sum of the series $\frac{1}{1+1^{2}+1^{4}}+\frac{2}{1+2^{2}+2^{4}}+\frac{3}{1+3^{2}+3^{4}}+\ldots$ to $n$ terms
is
(A) $\frac{n\left(n^{2}+1\right)}{n^{2}+n+1}$
(B) $\frac{n(n+1)}{2\left(n^{2}+n+1\right)}$
(C) $\frac{n\left(n^{2}-1\right)}{2\left(n^{2}+n+1\right)}$
(D) None of these

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a complicated rational function into a sum of simpler fractions. This method makes it easier to analyze and sum series, particularly when each term in the series has a rational expression that can be split into components that might cancel out in the overall sum.

Telescoping Series

A telescoping series is a series where most terms cancel out with neighboring terms when summed, leaving only a few terms that determine the final sum. Recognizing the telescoping nature after decomposing a term via partial fractions allows for the efficient computation of the series' sum without having to evaluate each term individually.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of lower-degree polynomials. In problems like this, identifying a factorization pattern, such as decomposing a quartic polynomial into the product of quadratics, can simplify complex expressions and reveal opportunities for further algebraic manipulations.

Series Summation

Series summation involves finding a closed-form expression for the sum of a sequence of terms. Through algebraic manipulation and recognizing patterns like telescoping behavior, one can transform a complex series into an expression that sums to a finite, simplified form.

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