The sum of the series 2-1- - 6-2- - 12-3- - 20-4- - - is

The correct option is D 3e Writing the general term, we get T_n=n(n+1)n! =n+1(n 1)! Reconstructing the series, we get S=2+3+42!+53!...∞ ...

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Byju's Answer

Standard XII

Mathematics

Logarithmic Function

The sum of th...

Question

The sum of the series $\frac{2}{1!} + \frac{6}{2!} + \frac{12}{3!} + \frac{20}{4!} + . . .$ is

\frac{3 e}{2}

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e

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2 e

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3 e

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Solution

The correct option is D $3 e$
Writing the general term, we get
$T_{n} = \frac{n (n + 1)}{n!}$
$= \frac{n + 1}{(n - 1)!}$
Reconstructing the series, we get
$S = 2 + 3 + \frac{4}{2!} + \frac{5}{3!} . . . \infty$
$= {\frac{d (x^{2} e^{x})}{d x}}_{x = 1}$
$= [2 x e^{x} + x^{2} e^{x}]_{x = 1}$
$= 2 e + e$
$= 3 e$

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Similar questions

Q. In polymeric

(B e C l_{2})_{n}

, there are:

Q. Find the

△ G^{\circ}

of a galvanic cell with

A g

and

A l

Standard Reduction Potentials at

25^{\circ} C

Half-Reaction	$E^{\circ} (V)$
$F_{2} + 2 e^{-} \to 2 F^{-}$	$2.87$
$A g^{2} + e^{-} \to A g^{+}$	$1.99$
$C o^{3 +} + e^{-} \to C o^{2 +}$	$1.82$
$H_{2} O_{2} + 2 H^{+} + 2 e^{-} \to 2 H_{2} O$	$1.78$
$M n O_{4}^{-} + 4 H^{+} + 3 e^{-} \to M n O_{2} + 2 H_{2} O$	$1.68$
$2 e^{-} + 2 H^{+} + I O_{4}^{-} \to I O_{3}^{-} + H_{2} O$	$1.60$
$M n O_{4}^{-} + 8 H^{+} + 5 e^{-} \to M n^{2 +} + 4 H_{2} O$	$1.51$
$A u^{3 +} + 3 e^{-} \to A u$	$1.50$
$C l_{2} + 2 e^{-} \to 2 C l^{-}$	$1.36$
$O_{2} + 4 H^{+} + 4 e^{-} \to 2 H_{2} O$	$1.23$
$M n O_{2} + 4 H^{+} + 2 e^{-} \to M n^{2 +} + 2 H_{2} O$	$1.21$
$B r_{2} + 2 e^{-} \to 2 B r^{-}$	$1.09$
$N O_{3}^{-} + 4 H^{+} + 3 e^{-} \to N O + 2 H_{2} O$	$0.96$
$C l O_{2} + e^{-} \to C l O_{2}$	$0.954$
$2 H g^{2 +} + 2 e^{-} \to H g_{2}^{2 +}$	$0.91$
$A g^{+} + e^{-} \to A g$	$0.80$
$H g_{2}^{2 +} + 2 e^{-} \to 2 H g$	$0.80$
$F e^{3 +} + e^{-} \to F e^{2 +}$	$0.77$
$O_{2} + 2 H^{+} + 2 e^{-} \to H_{2} O_{2}$	$0.68$
$M n O_{4}^{-} + e^{-} \to M n O_{4}^{2 -}$	$0.56$
$I_{2} + 2 e^{-} \to 2 I^{-}$	$0.54$
$C u^{+} + e^{-} \to C u$	$0.52$
$O_{2} + 2 H_{2} O + 4 e^{-} \to 4 O H^{-}$	$0.40$
$C u^{2 +} + 2 e^{-} \to C u$	$0.34$
$A g C l + e^{-} \to A g + C l^{-}$	$0.22$
$C u^{2 +} + e^{-} \to C u^{+}$	$0.16$
$2 H^{+} + 2 e^{-} \to H_{2}$	$0.00$
$F e^{3 +} + 3 e^{-} \to F e$	$- 0.036$
$P b^{2 +} + 2 e^{-} \to P b$	$- 0.13$
$N i^{2 +} + 2 e^{-} \to N i$	$- 0.23$
$C d^{2 +} + 2 e^{-} \to C d$	$- 0.40$
$F e^{2 +} + 2 e^{-} \to F e$	$- 0.44$
$C r^{3 +} + e^{-} \to C r^{2 +}$	$- 0.50$
$C r^{3 +} + 3 e^{-} \to C r$	$- 0.73$
$Z n^{2 +} + 2 e^{-} \to Z n$	$- 0.76$
$2 H_{2} O + 2 e^{-} \to H_{2} + 2 O H^{-}$	$- 0.83$
$M n^{2 +} + 2 e^{-} \to M n$	$- 1.18$
$A l^{3 +} + 3 e^{-} \to A l$	$- 1.66$
$H_{2} + 2 e^{-} \to 2 H^{-}$	$- 2.23$
$M g^{2 +} + 2 e^{-} \to M g$	$- 2.37$
$N a^{+} + e^{-} \to N a$	$- 2.71$
$C a^{2 +} + 2 e^{-} \to C a$	$- 2.76$
$B a^{2 +} + 2 e^{-} \to B a$	$- 2.90$
$K^{+} + e^{-} \to K$	$- 2.92$
$L i^{+} + e^{-} \to L i$	$- 3.05$

The oxidation and reduction half reactions of the reaction

2Al_(s) + 3Cu²⁺_(aq) → 2Al³⁺_(aq) + 3Cu_(s) are

Q. Rank the following elements by increasing ease of oxidation with most easily oxidized LAST:

C d, C u, N i, F e

Standard Reduction Potentials at

25^{\circ} C

Half-Reaction	$E^{\circ} (V)$
$F_{2} + 2 e^{-} \to 2 F^{-}$	$2.87$
$A g^{2} + e^{-} \to A g^{+}$	$1.99$
$C o^{3 +} + e^{-} \to C o^{2 +}$	$1.82$
$H_{2} O_{2} + 2 H^{+} + 2 e^{-} \to 2 H_{2} O$	$1.78$
$M n O_{4}^{-} + 4 H^{+} + 3 e^{-} \to M n O_{2} + 2 H_{2} O$	$1.68$
$2 e^{-} + 2 H^{+} + I O_{4}^{-} \to I O_{3}^{-} + H_{2} O$	$1.60$
$M n O_{4}^{-} + 8 H^{+} + 5 e^{-} \to M n^{2 +} + 4 H_{2} O$	$1.51$
$A u^{3 +} + 3 e^{-} \to A u$	$1.50$
$C l_{2} + 2 e^{-} \to 2 C l^{-}$	$1.36$
$O_{2} + 4 H^{+} + 4 e^{-} \to 2 H_{2} O$	$1.23$
$M n O_{2} + 4 H^{+} + 2 e^{-} \to M n^{2 +} + 2 H_{2} O$	$1.21$
$B r_{2} + 2 e^{-} \to 2 B r^{-}$	$1.09$
$N O_{3}^{-} + 4 H^{+} + 3 e^{-} \to N O + 2 H_{2} O$	$0.96$
$C l O_{2} + e^{-} \to C l O_{2}$	$0.954$
$2 H g^{2 +} + 2 e^{-} \to H g_{2}^{2 +}$	$0.91$
$A g^{+} + e^{-} \to A g$	$0.80$
$H g_{2}^{2 +} + 2 e^{-} \to 2 H g$	$0.80$
$F e^{3 +} + e^{-} \to F e^{2 +}$	$0.77$
$O_{2} + 2 H^{+} + 2 e^{-} \to H_{2} O_{2}$	$0.68$
$M n O_{4}^{-} + e^{-} \to M n O_{4}^{2 -}$	$0.56$
$I_{2} + 2 e^{-} \to 2 I^{-}$	$0.54$
$C u^{+} + e^{-} \to C u$	$0.52$
$O_{2} + 2 H_{2} O + 4 e^{-} \to 4 O H^{-}$	$0.40$
$C u^{2 +} + 2 e^{-} \to C u$	$0.34$
$A g C l + e^{-} \to A g + C l^{-}$	$0.22$
$C u^{2 +} + e^{-} \to C u^{+}$	$0.16$
$2 H^{+} + 2 e^{-} \to H_{2}$	$0.00$
$F e^{3 +} + 3 e^{-} \to F e$	$- 0.036$
$P b^{2 +} + 2 e^{-} \to P b$	$- 0.13$
$N i^{2 +} + 2 e^{-} \to N i$	$- 0.23$
$C d^{2 +} + 2 e^{-} \to C d$	$- 0.40$
$F e^{2 +} + 2 e^{-} \to F e$	$- 0.44$
$C r^{3 +} + e^{-} \to C r^{2 +}$	$- 0.50$
$C r^{3 +} + 3 e^{-} \to C r$	$- 0.73$
$Z n^{2 +} + 2 e^{-} \to Z n$	$- 0.76$
$2 H_{2} O + 2 e^{-} \to H_{2} + 2 O H^{-}$	$- 0.83$
$M n^{2 +} + 2 e^{-} \to M n$	$- 1.18$
$A l^{3 +} + 3 e^{-} \to A l$	$- 1.66$
$H_{2} + 2 e^{-} \to 2 H^{-}$	$- 2.23$
$M g^{2 +} + 2 e^{-} \to M g$	$- 2.37$
$N a^{+} + e^{-} \to N a$	$- 2.71$
$C a^{2 +} + 2 e^{-} \to C a$	$- 2.76$
$B a^{2 +} + 2 e^{-} \to B a$	$- 2.90$
$K^{+} + e^{-} \to K$	$- 2.92$
$L i^{+} + e^{-} \to L i$	$- 3.05$

E^{⊖} f o r C r^{3 +} + 3 e^{-} \to C r a n d C r^{3 +} + e^{-} \to C r^{2 +}

are

- 0.74 V

and

- 0.40 V

, respectively.

E^{⊖}

for the reaction is:

C r^{+ 2} + 2 e^{-} \to C r

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The sum of the series 21!+62!+123!+204!+... is

The correct option is D 3eWriting the general term, we getTn=n(n+1)n!=n+1(n−1)!Reconstructing the series, we getS=2+3+42!+53!...∞=d(x2ex)dxx=1=[2xex+x2ex]x=1=2e+e=3e

The sum of the series $\frac{2}{1!} + \frac{6}{2!} + \frac{12}{3!} + \frac{20}{4!} + . . .$ is