lim-n-1-n-1-2-1-n-1-2-2-n-1-2-3-n-1-2-n-n-

Question Number 196471 by Tawa11 last updated on 25/Aug/23

\lim_{n \to \infty} \frac{1}{n} (\frac{1}{2 + \frac{1}{n}} + \frac{1}{2 + \frac{2}{n}} + \frac{1}{2 + \frac{3}{n}} + . . + \frac{1}{2 + \frac{n}{n}})

Answered by mr W last updated on 25/Aug/23

= \int_{0}^{1} \frac{d x}{2 + x}

= {[\ln (2 + x)]}_{0}^{1}

= \ln 3 - \ln 2

= \ln \frac{3}{2}

Commented by Tawa11 last updated on 25/Aug/23

God bless you sir . I appreciate .

Commented by Tawa11 last updated on 25/Aug/23

Sir is the sum a general term I should know ?

Or you solved to get \int_{0}^{1} \frac{1}{2 + x} dx

Commented by mr W last updated on 25/Aug/23

\int_{0}^{1} \frac{1}{2 + x} d x

= \lim_{Δ x \to 0} Σ \frac{1}{2 + x_{k}} \times Δ x

= \lim_{n \to \infty} \underset{k = 1}{\sum^{n}} \frac{1}{2 + \frac{k}{n}} \times \frac{1}{n}

= \lim_{n \to \infty} (\frac{1}{n} \underset{k = 1}{\sum^{n}} \frac{1}{2 + \frac{k}{n}})

Commented by mr W last updated on 25/Aug/23

Commented by mr W last updated on 25/Aug/23

g e n e r a l l y

Δ x = \frac{1}{n}

\int_{0}^{1} f (x) d x \approx y_{1} Δ x + y_{2} Δ x + \dots + y_{k} Δ x + \dots + y_{n} Δ x

\int_{0}^{1} f (x) d x \approx (y_{1} + y_{2} + \dots + y_{k} + \dots + y_{n}) Δ x

\int_{0}^{1} f (x) d x \approx [f (x_{1}) + f (x_{2}) + \dots + f (x_{k}) + \dots + f (x_{n})] Δ x

\int_{0}^{1} f (x) d x \approx [f (Δ x) + f (2 Δ x) + \dots + f (k Δ x) + \dots + f (n Δ x)] Δ x

\int_{0}^{1} f (x) d x \approx [f (\frac{1}{n}) + f (\frac{2}{b}) + \dots + f (\frac{k}{n}) + \dots + f (\frac{n}{n})] \frac{1}{n}

\int_{0}^{1} f (x) d x \approx \frac{1}{n} \underset{k = 1}{\sum^{n}} f (\frac{k}{n})

\int_{0}^{1} f (x) d x = \lim_{n \to \infty} [\frac{1}{n} \underset{k = 1}{\sum^{n}} f (\frac{k}{n})]

i n c u r r e n t c a s e : f (x) = \frac{1}{2 + x}

Commented by Tawa11 last updated on 25/Aug/23

Wow, I really appreciate sir .

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