The value of100limn- 200-C1-1-12C2-1-12-13C3- -1n-11-12-13-1nCn- is where Cr-nCr

Let T=[C_1 (1+12)C_2+(1+12+13)C_3 ⋯ +( 1)^n 1(1+12+13+⋯+1n)C_n] The r^th term of the series is nbsp; T(r)=( 1)^r 1(1+12+13+⋯+1r)C_r Observe that T(r)=∫_0^1 ...

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Byju's Answer

Standard XII

Mathematics

Integration to Solve Modified Sum of Binomial Coefficients

The value of1...

Question

The value of
$100 lim n \to 200 [C_{1} - (1 + \frac{1}{2}) C_{2} + (1 + \frac{1}{2} + \frac{1}{3}) C_{3} - \dots + (- 1)^{n - 1} (1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}) C_{n}]$ is (where $C_{r} =^{n} C_{r}$ )

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Solution

Let $T = [C_{1} - (1 + \frac{1}{2}) C_{2} + (1 + \frac{1}{2} + \frac{1}{3}) C_{3} - \dots + (- 1)^{n - 1} (1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}) C_{n}]$

The $r^{t h}$ term of the series is
$T (r) = (- 1)^{r - 1} (1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{r}) C_{r}$

Observe that $T (r) = 1 \int 0 (- 1)^{r - 1} C_{r} (1 + x + x^{2} + \dots + x^{r - 1}) d x$

So consider a series whose $r^{t h}$ term is
$T_{1} (r)$ , where
$T_{1} (r) = (- 1)^{r - 1} C_{r} (1 + x + x^{2} + \dots + x^{r - 1}) = (- 1)^{r - 1} C_{r} \frac{1 - x^{r}}{1 - x} = \frac{(- 1)^{r} C_{r}}{x - 1} - \frac{(- 1)^{r} x^{r} C_{r}}{x - 1}$

$\Rightarrow n \sum r = 1 T_{1} (r) = \frac{1}{(x - 1)} [n \sum r = 1 (- 1)^{r} C_{r} - n \sum r = 1 (- 1)^{r} C_{r} x^{r}] = \frac{1}{x - 1} (1 - 1)^{n} + \frac{1}{1 - x} (1 - x)^{n} = (1 - x)^{n - 1}$
$∴$ Sum, $S = 1 \int 0 (1 - x)^{n - 1} d x = \frac{1}{n}$
$∴ 100 lim n \to 200 (S) = 0.5$

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The value of 100limn→200[C1−(1+12)C2+(1+12+13)C3−⋯+(−1)n−1(1+12+13+⋯+1n)Cn] is (where Cr=nCr)

The value of
$100 lim n \to 200 [C_{1} - (1 + \frac{1}{2}) C_{2} + (1 + \frac{1}{2} + \frac{1}{3}) C_{3} - \dots + (- 1)^{n - 1} (1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}) C_{n}]$ is (where $C_{r} =^{n} C_{r}$ )