Prove: $ b_n underset{ntoinfty}{longrightarrow} L, text{as} b_n = frac{sum_{k=1}^{n}t_{k}cdot...
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Given $(a_{n})_{n=1}^infty$ , a sequence that converges to $L$ ($L$ is not necessarily $in mathbb{R}$).
Let $left(t_{n}right)$ be a positive sequence such that:
$$ sum_{k=1}^{n}t_{k} underset{ntoinfty}{longrightarrow} infty$$
We'll define:
$$ b_n = frac{sum_{k=1}^{n}t_{k}cdot {a_k}}{sum_{k=1}^{n}t_{k}} $$
Prove: $ b_n underset{ntoinfty}{longrightarrow} L$.
calculus sequences-and-series
asked Nov 15 at 13:06
Jneven
684320
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up vote
0
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Given $(a_{n})_{n=1}^infty$ , a sequence that converges to $L$ ($L$ is not necessarily $in mathbb{R}$).
Let $left(t_{n}right)$ be a positive sequence such that:
$$ sum_{k=1}^{n}t_{k} underset{ntoinfty}{longrightarrow} infty$$
We'll define:
$$ b_n = frac{sum_{k=1}^{n}t_{k}cdot {a_k}}{sum_{k=1}^{n}t_{k}} $$
Prove: $ b_n underset{ntoinfty}{longrightarrow} L$.
calculus sequences-and-series
asked Nov 15 at 13:06
Jneven
684320
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given $(a_{n})_{n=1}^infty$ , a sequence that converges to $L$ ($L$ is not necessarily $in mathbb{R}$).
Let $left(t_{n}right)$ be a positive sequence such that:
$$ sum_{k=1}^{n}t_{k} underset{ntoinfty}{longrightarrow} infty$$
We'll define:
$$ b_n = frac{sum_{k=1}^{n}t_{k}cdot {a_k}}{sum_{k=1}^{n}t_{k}} $$
Prove: $ b_n underset{ntoinfty}{longrightarrow} L$.
calculus sequences-and-series
asked Nov 15 at 13:06
Jneven
684320
Given $(a_{n})_{n=1}^infty$ , a sequence that converges to $L$ ($L$ is not necessarily $in mathbb{R}$).
Let $left(t_{n}right)$ be a positive sequence such that:
$$ sum_{k=1}^{n}t_{k} underset{ntoinfty}{longrightarrow} infty$$
We'll define:
$$ b_n = frac{sum_{k=1}^{n}t_{k}cdot {a_k}}{sum_{k=1}^{n}t_{k}} $$
Prove: $ b_n underset{ntoinfty}{longrightarrow} L$.
calculus sequences-and-series
calculus sequences-and-series
asked Nov 15 at 13:06
Jneven
684320
asked Nov 15 at 13:06
Jneven
684320
edited Nov 16 at 12:12
asked Nov 15 at 13:06
Jneven
684320
asked Nov 15 at 13:06
Jneven
684320
asked Nov 15 at 13:06
Jneven
684320
684320
add a comment |
add a comment |
1 Answer
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It is an easy consquence of Theorem of Stolz : https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem
answered Nov 16 at 12:09
MotylaNogaTomkaMazura
6,311917
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
It is an easy consquence of Theorem of Stolz : https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem
answered Nov 16 at 12:09
MotylaNogaTomkaMazura
6,311917
add a comment |
up vote
2
down vote
accepted
It is an easy consquence of Theorem of Stolz : https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem
answered Nov 16 at 12:09
MotylaNogaTomkaMazura
6,311917
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
It is an easy consquence of Theorem of Stolz : https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem
answered Nov 16 at 12:09
MotylaNogaTomkaMazura
6,311917
It is an easy consquence of Theorem of Stolz : https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem
answered Nov 16 at 12:09
MotylaNogaTomkaMazura
6,311917
answered Nov 16 at 12:09
MotylaNogaTomkaMazura
6,311917
answered Nov 16 at 12:09
MotylaNogaTomkaMazura
6,311917
answered Nov 16 at 12:09
MotylaNogaTomkaMazura
6,311917
6,311917
add a comment |
add a comment |
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