Let `l _(n) =int _(-1) ^(1) |x|(1+ x+ (x ^(2))/(2 ) +(x ^(2))/(3) + ..... + (x ^(2n))/(2n))dx if lim _(x to oo) l _(n) ` can be expressed as rational `p/q` in this lowest form, then find the value of `(pq(p+q))/(10)`

Answer

To solve the problem, we need to evaluate the integral

\[ l_n = \int_{-1}^{1} |x| \left(1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \ldots + \frac{x^{2n}}{2n}\right) dx \]

and find the limit as \( n \to \infty \).

### Step 1: Simplifying the Integral

The expression inside the integral is a series expansion. We can rewrite it as:

\[ l_n = \int_{-1}^{1} |x| \left( \sum_{k=0}^{2n} \frac{x^k}{k} \right) dx \]

### Step 2: Understanding the Even and Odd Functions

Since \( |x| \) is an even function and the series contains both even and odd powers of \( x \), we can separate the integral into two parts. The odd powers will integrate to zero over the symmetric interval \([-1, 1]\).

Thus, we only need to consider the even powers:

\[ l_n = 2 \int_{0}^{1} x \left( 1 + \frac{x^2}{2} + \frac{x^4}{4} + \ldots + \frac{x^{2n}}{2n} \right) dx \]

### Step 3: Evaluating the Integral

Now we can focus on the integral:

\[ l_n = 2 \int_{0}^{1} x \left( 1 + \frac{x^2}{2} + \frac{x^4}{4} + \ldots + \frac{x^{2n}}{2n} \right) dx \]

This can be expressed as:

\[ = 2 \left( \int_{0}^{1} x dx + \frac{1}{2} \int_{0}^{1} x^3 dx + \frac{1}{4} \int_{0}^{1} x^5 dx + \ldots + \frac{1}{2n} \int_{0}^{1} x^{2n+1} dx \right) \]

### Step 4: Calculating Each Integral

The integral \( \int_{0}^{1} x^k dx = \frac{1}{k+1} \). Therefore, we have:

\[ \int_{0}^{1} x^{2k+1} dx = \frac{1}{2k+2} \]

Substituting this back into our expression gives:

\[ l_n = 2 \left( 1 + \frac{1}{2(3)} + \frac{1}{4(5)} + \ldots + \frac{1}{2n(2n+1)} \right) \]

### Step 5: Finding the Limit as \( n \to \infty \)

As \( n \to \infty \), the series converges to:

\[ \lim_{n \to \infty} l_n = 2 \left( 1 + \frac{1}{2} + \frac{1}{4} + \ldots \right) \]

This series converges to:

\[ = 2 \left( 1 + \frac{1}{2} + \frac{1}{4} + \ldots \right) = 2 \cdot 2 = 4 \]

### Step 6: Expressing in the Form \( \frac{p}{q} \)

We have \( \lim_{n \to \infty} l_n = \frac{4}{1} \). Here, \( p = 4 \) and \( q = 1 \).

### Step 7: Finding \( \frac{pq(p+q)}{10} \)

Now we calculate:

\[ pq(p + q) = 4 \cdot 1 \cdot (4 + 1) = 4 \cdot 1 \cdot 5 = 20 \]

Finally, we find:

\[ \frac{pq(p + q)}{10} = \frac{20}{10} = 2 \]

### Final Answer

Thus, the answer is:

\[ \boxed{2} \]