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Question Number 66620 by Mohamed Amine Bouguezzoul last updated on 18/Aug/19
find lim_(n→∞)  I_n   I_n =∫_0 ^∞ (dx/((1+coth (nx))^n )) ,n≥1
findlimnInIn=0dx(1+coth(nx))n,n1
Commented by mathmax by abdo last updated on 19/Aug/19
coth(nx) =((ch(nx))/(sh(nx))) =((e^(nx)  +e^(−nx) )/(e^(nx) −e^(−nx) ))  ⇒1+coth(nx) =1+((e^(nx)  +e^(−nx) )/(e^(nx) −e^(−nx) ))  =((e^(nx) −e^(−nx)  +e^(nx) +e^(−nx) )/(e^(nx) −e^(−nx) )) =((2e^(nx) )/(e^(nx) (1−e^(−2nx) ))) =(2/(1−e^(−2nx) )) ⇒  I_n =∫_0 ^∞      (dx/(((2/(1−e^(−2nx) )))^n )) =∫_0 ^∞   (1/2^n )(1−e^(−nx) )^n  dx  =∫_0 ^∞  (1/2^n )e^(nln(1−e^(−nx) )) dx  =∫_0 ^∞ f_n (x)dx with f_n (x) =(1/2^n )e^(nln(1−e^(−nx) ))   we have ln(1−e^(−nx) )∼−e^(−nx)  and nln(1−e^(−nx) )∼−ne^(−nx)  ⇒  f_n (x)∼(1/2^n )e^(−ne^(−nx) )  →0 (n→+∞)   the sequences (f_n )are continues  lim_(n→+∞)  I_n =∫_R^+   lim_(n→+∞)    f_n (x) =0
coth(nx)=ch(nx)sh(nx)=enx+enxenxenx1+coth(nx)=1+enx+enxenxenx=enxenx+enx+enxenxenx=2enxenx(1e2nx)=21e2nxIn=0dx(21e2nx)n=012n(1enx)ndx=012nenln(1enx)dx=0fn(x)dxwithfn(x)=12nenln(1enx)wehaveln(1enx)enxandnln(1enx)nenxfn(x)12nenenx0(n+)thesequences(fn)arecontinueslimn+In=R+limn+fn(x)=0
Commented by Mohamed Amine Bouguezzoul last updated on 20/Aug/19
excellent work abdo.
excellentworkabdo.
Commented by mathmax by abdo last updated on 26/Aug/19
you are welcome.
youarewelcome.

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