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lim-x-1-x-1-1-x-2-1-2x-




Question Number 187983 by TUN last updated on 24/Feb/23
lim_(x→∞)  ((1/(x+1))+(1/(x+2))+...+(1/(2x)))
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{{x}+\mathrm{1}}+\frac{\mathrm{1}}{{x}+\mathrm{2}}+…+\frac{\mathrm{1}}{\mathrm{2}{x}}\right) \\ $$
Answered by cortano12 last updated on 24/Feb/23
= lim_(x→∞)  Σ_(k=1) ^n ((1/(x+k)))  = lim_(x→∞)  (1/x)Σ_(k=1) ^n ((1/(1+(k/x))) )  = ∫_0 ^1 ((1/(1+x)))dx =ln 2
$$=\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{x}+\mathrm{k}}\right) \\ $$$$=\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{x}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{k}}{\mathrm{x}}}\:\right) \\ $$$$=\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\left(\frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}}\right)\mathrm{dx}\:=\mathrm{ln}\:\mathrm{2}\: \\ $$
Commented by Ar Brandon last updated on 24/Feb/23
Σ_(k=1) ^x
$$\underset{{k}=\mathrm{1}} {\overset{{x}} {\sum}} \\ $$

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