Show that 1 1 2 - 1 2 2 - - 1 n-1 2 - 1 1- 1 4 1 2 - 2 2 - - n 4 n 2 - n-1 2-

Let 1 u _ r ^ 2 + 1 u _ r+1 ^ 2 = 1 u _ r ^ 2 + x _ r ∴ x _ r = u _ r ^ 4 u _ r ^ 2 + u _ r+1 ^ 2 So, we get the result by putting; u _ r ...

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Byju's Answer

Standard XII

Mathematics

Arithmetic Mean

Show that 1 ...

Question

Show that
$\frac{1}{1^{2}} + \frac{1}{2^{2}} + \dots + \frac{1}{{(n + 1)}^{2}} = \frac{1}{1 -} \frac{1^{4}}{1^{2} + 2^{2} -} \dots \frac{n^{4}}{n^{2} + {(n + 1)}^{2}}$ .

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Solution

Let $\frac{1}{u_{r}^{2}} + \frac{1}{u_{r + 1}^{2}} = \frac{1}{u_{r}^{2} + x_{r}}$
$∴ x_{r} = - \frac{u_{r}^{4}}{u_{r}^{2} + u_{r + 1}^{2}}$
So, we get the result by putting;-
$u_{r} = r$
$\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + . . . . + \frac{1}{{(n + 1)}^{2}} = \frac{1}{1 -} \frac{1^{4}}{1^{2} + 2^{2} -} \frac{2^{4}}{2^{2} + 3^{2} -} . . . . . . \frac{n^{4}}{n^{2} + {(n + 1)}^{2} -}$

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Similar questions

Q. Show that

\frac{1 \times 2^{2} + 2 \times 3^{2} + . . . + n \times {(n + 1)}^{2}}{1^{2} \times 2 + 2^{2} \times 3 + . . . + n^{2} \times (n + 1)} = \frac{3 n + 5}{3 n + 1}

Show that

$\frac{1 \times 2^{2} + 2 \times 3^{2} + \dots \dots + n \times (n + 1)^{2}}{1^{2} \times 2 + 2^{2} \times 3 + \dots \dots + n^{2} \times (n + 1)} = \frac{3 n + 5}{3 n + 1}$

Q. Evaluate:

lim n \to \infty \frac{1}{n^{4}} [1^{2} + (1^{2} + 2^{2}) + (1^{2} + 2^{2} + 3^{2}) + . . . . . . (1^{2} + 2^{2} + 3^{2} + . . . . n^{2})]

{lim}_{n \to \infty} ⎡ ⎢ ⎢ ⎣ \frac{1}{n} + \frac{1}{\sqrt{n^{2} - 1^{2}}} + \frac{1}{\sqrt{n^{2} - 2^{2}}} + . . . . + \frac{1}{\sqrt{n^{2} - {(n - 1)}^{2}}} ⎤ ⎥ ⎥ ⎦

Q. Solve

lim n \to \infty {\frac{1}{n} + \frac{1}{\sqrt{n^{2} - 1^{2}}} + \frac{1}{\sqrt{n^{2} - 2^{2}}} + . . . + \frac{1}{\sqrt{n^{2} - {(n - 1)}^{2}}}}

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Show that112+122+⋯+1(n+1)2=11−1412+22−⋯n4n2+(n+1)2.

Let 1u2r+1u2r+1=1u2r+xr∴xr=−u4ru2r+u2r+1So, we get the result by putting;-ur=r112+122+132+....+1(n+1)2=11−1412+22−2422+32−......n4n2+(n+1)2−

Show that
$\frac{1}{1^{2}} + \frac{1}{2^{2}} + \dots + \frac{1}{{(n + 1)}^{2}} = \frac{1}{1 -} \frac{1^{4}}{1^{2} + 2^{2} -} \dots \frac{n^{4}}{n^{2} + {(n + 1)}^{2}}$ .