Question

Express the limits as definite integrals. \lim_{||P||\rightarrow 0}\sum_{k=1}^{n}(\frac{1}{C_{k}})\triangle x_{k}, where P is a partition of [1,4]

Applications of integrals
ANSWERED
asked 2021-05-17

Express the limits as definite integrals. \(\displaystyle\lim_{{{P}\rightarrow{0}}}{\sum_{{{k}={1}}}^{{{n}}}}{\left({\frac{{{1}}}{{{C}_{{{k}}}}}}\right)}\triangle{x}_{{{k}}}\), where P is a partition of [1,4]

Answers (1)

2021-05-18
Step 1
Given
\(\lim_{||P||\rightarrow 0}\sum_{k=1}^{n}(\frac{1}{C_{k}})\triangle x_{k}\)
Step 2
To express limits as a definite integrals.
The definition of definite integral is,
\(\int_{a}^{b}f(x)dx=\lim_{n\rightarrow \infty}\sum_{i=1}^{n}f(x_{i})\triangle x\)
Here \(f(x_{k})=\frac{1}{C_{k}}\)
And p is a partition of [1,4],
Therefore,
\(\lim_{n\rightarrow \infty}\sum_{k=1}^{n}(\frac{1}{C_{k}})\triangle x_{k}=\int_{1}^{4}f(x)dx\)
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