Answer

Verified

426.6k+ views

**Hint:**

We are given the general formula of ${r^{th}}$ term. Substituting different values for $r$ and comparing with the given terms we get the value of $a, b$ and $c$. Then we can consider the partial sum of the expression and see the infinite sum by tending $n$ to infinity.

**Useful formula:**

For any $a,b$ we have the algebraic identities.

${(a + b)^2} = {a^2} + 2ab + {b^2}$

${a^2} - {b^2} = (a - b)(a + b)$

**Complete step by step solution:**

It is given that the \[{r^{th}}\] term of the series is $\dfrac{{ar}}{{b{r^4} + c}}$

So substituting $r = 1$, we get the first term.

That is, $\dfrac{{a \times 1}}{{b \times {1^4} + c}} = \dfrac{a}{{b + c}}$

But we have, first term given as $\dfrac{4}{5}$.

This gives, $\dfrac{a}{{b + c}} = \dfrac{4}{5}$.

So we have, $a = 4,b + c = 5$

Now substitute $r = 2$,

$\dfrac{{a \times 2}}{{b \times {2^4} + c}} = \dfrac{{2a}}{{16b + c}}$

But we have, second term as $\dfrac{8}{{65}}$.

Comparing we get,

$16b + c = 65$

Also $b + c = 5$

Subtracting both sides we get,

$15b = 60$

Dividing both sides by $15$ we get,

$b = \dfrac{{60}}{{15}} = 4$

$ \Rightarrow c = 5 - b = 5 - 4 = 1$

So we have, $a = 4,b = 4,c = 1$

Therefore the ${r^{th}}$ term becomes $\dfrac{{4r}}{{4{r^4} + 1}}$.

Consider $4{r^4} + 1$.

Adding and subtracting $4{r^2}$ in the right side we get,

$4{r^4} + 1 = 4{r^4} + 4{r^2} + 1 - 4{r^2}$

Now we have,

$4{r^4} + 1 = (4{r^4} + 4{r^2} + 1) - 4{r^2}$

We have ${(a + b)^2} = {a^2} + 2ab + {b^2}$

Using this we get,

$4{r^4} + 1 = {(2{r^2} + 1)^2} - 4{r^2}$

We know, ${a^2} - {b^2} = (a - b)(a + b)$

$ \Rightarrow 4{r^4} + 1 = [(2{r^2} + 1) - 2r][(2{r^2} + 1) + 2r]$

$ \Rightarrow 4{r^4} + 1 = (2{r^2} + 1 - 2r)(2{r^2} + 1 + 2r)$

Then $\dfrac{{4r}}{{4{r^4} + 1}} = \dfrac{{4r}}{{(2{r^2} + 1 - 2r)(2{r^2} + 1 + 2r)}}$

$\dfrac{{4r}}{{4{r^4} + 1}} = \dfrac{1}{{(2{r^2} + 1 - 2r)}} - \dfrac{1}{{(2{r^2} + 1 + 2r)}}$

Now consider the partial sum,

$\sum\limits_1^n {\dfrac{{4r}}{{4{r^4} + 1}}} = \sum\limits_1^n {(\dfrac{1}{{(2{r^2} + 1 - 2r)}} - \dfrac{1}{{(2{r^2} + 1 + 2r)}}} )$

Substituting values $r = 1,2,3...$ we get,

\[\sum\limits_1^n {\dfrac{{4r}}{{4{r^4} + 1}}} = \dfrac{1}{{2 + 1 - 2}} - \dfrac{1}{{2 + 1 + 2}} + \dfrac{1}{{2 + 1 + 2}} - \dfrac{1}{{8 + 1 + 4}} + \dfrac{1}{{8 + 1 + 4}} - ...\]

\[\sum\limits_1^n {\dfrac{{4r}}{{4{r^4} + 1}}} = 1 - \dfrac{1}{5} + \dfrac{1}{5} - \dfrac{1}{{13}} + \dfrac{1}{{13}} - ...\]

So we can see that the consecutive terms get cancelling.

When $n \to \infty $, \[\sum\limits_1^n {\dfrac{{4r}}{{4{r^4} + 1}}} = 1 - \dfrac{1}{5} + \dfrac{1}{5} - \dfrac{1}{{13}} + \dfrac{1}{{13}} - ... \to 1\]

So we get the sum up to infinity of the series is $1$.

**$\therefore $ The answer is option D.**

**Note:**

This problem includes a lot of steps. So we may have made mistakes easily. Also remember that when $n \to \infty $, then $\dfrac{1}{n} \to 0$ and so $1 - \dfrac{1}{n} \to 1$. We often use the idea of partial sum in finding the infinite sum.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE