Which of the following is true for a series -0- an-

The correct option is E If lim_n→∞a_n ≠ 0 , then the series diverges.Given : ∑ _ n=0 ^∞ #160; a _ n Then, n→∞ #160; lim a≠ 0 then, the s ...

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Standard XII

Chemistry

Real Gases

Which of the ...

Question

Which of the following is true for a series $\infty \sum 0 a_{n}$ ?

If the series is convergent, then it is absolutely convergent.

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If the series is alternating, then it is convergent.

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lim n \to \infty a_{n} = L

, then

\infty \sum n = 1 a_{n} = L

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lim n \to \infty a_{n} = 0

, then the series converges.

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lim n \to \infty a_{n} \neq 0

, then the series diverges.

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Solution

The correct option is E If $lim n \to \infty a_{n} \neq 0$ , then the series diverges.
Given : $\sum_{n = 0}^{\infty} a_{n}$
Then, $l i m n \to \infty a \neq 0$ then, the series diverges.
Hence the correct answer is if $l i m n \to \infty a \neq 0$ then the series diverges.

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Similar questions

Q. Show that the infinite series

u_{1} + u_{2} + u_{3} + u_{4} + . . . . . .

is convergent or divergent according as

lim n \to \infty \sqrt[n]{u_{n}}

< 1

> 1

Q. If

n \to \infty

then the limit of series in

n

can be evaluated by following the rule :

lim n \to \infty c n + d \sum r = a n + b \frac{1}{n} f (\frac{r}{n}) = \int_{a}^{c} f (x) d x,

where in

L H S

\frac{r}{n}

is replaced by

x

\frac{1}{n}

d x

and the lower and upper limits are

{lim}_{n \to \infty} \frac{a n + b}{n} a n d {lim}_{n \to \infty} \frac{c n + d}{n}

respectively. Then answer the following question.

{lim}_{n \to \infty} {\frac{1}{\sqrt{4 n - 1^{2}}} + \frac{1}{\sqrt{8 n - 2^{2}}} + . . . + \frac{1}{\sqrt{3} n}}

equals?

A sequence is a function whose domain is the set of natural number. A sequence

s_{1}, s_{2} . . .

of real numbers is said to have a limit

l

{lim}_{n \to \infty} s_{n} = l

. If

l < \infty,

then

< s_{n} >

is said to convergent. The following are well known
(i)

lim n \to \infty \frac{1}{n^{p}} = 0 (p > 0)

(ii)

lim n \to \infty x^{n} = 0 (| x | < 1)

(iii) If

lim n \to \infty s_{n} = l

, then

lim n \to \infty \frac{s_{1} + s_{2} + . . . + s_{n}}{n} = l

Let

f (x) = - 1 + | x - 1 |, g (x) = 2 - | x + 1 |

then

Q. An

LR

series circuit has

L = 1 H and R = 1 Ω

. If it is connected across an emf of

2 V

then

Q. If sum of the series

\infty \sum n = 0 r^{n} = S

for

| r | < 1

, then sum of the series

\infty \sum n = 0 r^{2 n}

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Which of the following is true for a series ∞∑0an?

The correct option is E If limn→∞an≠0, then the series diverges.Given : ∑∞n=0anThen, limn→∞a≠0 then, the series diverges.Hence the correct answer is if limn→∞a≠0 then the series diverges.

Which of the following is true for a series $\infty \sum 0 a_{n}$ ?

The correct option is E If $lim n \to \infty a_{n} \neq 0$ , then the series diverges.
Given : $\sum_{n = 0}^{\infty} a_{n}$
Then, $l i m n \to \infty a \neq 0$ then, the series diverges.
Hence the correct answer is if $l i m n \to \infty a \neq 0$ then the series diverges.