Animated Solution for Mathematics - Sequence and Series: The sum to 10 terms of the series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\dots$ is:-

The series given is ${\sum }_{n=1}^{10}\frac{n}{n^4+n^2+1}$. At first glance, this expression appears to be a chaotic collection of powers and fractions.

However, in the context of JEE Advanced mathematics, such complexity is often order in disguise. Our first step is to isolate the general term:

The denominator $n^4+n^2+1$ is a classic algebraic structure. To simplify it, we employ the technique of completing the square by adding and subtracting $n^2$:

This transforms the expression into a difference of squares:

Applying the identity $a^2-b^2=(a-b)(a+b)$, we factorize the denominator into:

To utilize the Method of Differences, we express $T_n$ as a subtraction of two fractions. Observe the difference between the two factors:

Since our numerator is $n$, we multiply and divide by $2$ to match this difference:

Substituting the difference into the numerator, we obtain the golden form:

We now write out the first few terms to observe the cancellation pattern: For $n=1$: $T_1=\frac{1}{2}\left(\frac{1}{1}-\frac{1}{3}\right)$ For $n=2$: $T_2=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{7}\right)$ For $n=3$: $T_3=\frac{1}{2}\left(\frac{1}{7}-\frac{1}{13}\right)$

The negative term of each fraction cancels with the positive term of the subsequent fraction. This telescoping effect continues until the final term $T_{10}$:

When we sum all terms, every intermediate value is eliminated, leaving only the first and last components:

Simplifying the expression inside the parentheses:

The final result of the series is: