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CBSE Class 11-science Answered

Q.1Prove that log 2<1
Asked by Vidushi412 | 28 Nov, 2018, 07:15: PM
answered-by-expert Expert Answer
begin mathsize 12px style ln left parenthesis 1 plus x right parenthesis space equals space x space minus x squared over 2 plus x cubed over 3 minus x to the power of 4 over 4 plus x to the power of 5 over 5 minus...................... left parenthesis 1 right parenthesis end style   when   -1<x≤1
in Eqn.(1), put x = 1, then we have ,  begin mathsize 12px style ln space 2 space equals space 1 space minus space 1 half space plus 1 third minus 1 fourth...................... space left parenthesis 2 right parenthesis
end style
eqn.(2) can be written as,   begin mathsize 12px style l n 2 space equals space 1 space minus space open parentheses 1 half minus 1 third close parentheses minus open parentheses 1 fourth minus 1 fifth close parentheses......................... end style(3)
 
begin mathsize 12px style ln 2 space equals space 1 space minus space sum from n equals 1 to infinity of fraction numerator 1 over denominator open parentheses 2 n plus 1 close parentheses open parentheses 2 n close parentheses end fraction space..................... left parenthesis 5 right parenthesis end style
In eqn.(5), RHS is having a convergent series and its sum <1. hence ln2 <1
 
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begin mathsize 12px style ln space x space equals space 2 open curly brackets open parentheses fraction numerator x minus 1 over denominator x plus 1 end fraction close parentheses plus 1 third open parentheses fraction numerator x minus 1 over denominator x plus 1 end fraction close parentheses cubed plus 1 fifth open parentheses fraction numerator x minus 1 over denominator x plus 1 end fraction close parentheses to the power of 7............. close curly brackets
l e t space x space equals space 3
ln space 3 space equals space 2 space open curly brackets 1 half plus 1 third open parentheses 1 half close parentheses cubed plus 1 fifth open parentheses 1 half close parentheses to the power of 5 plus 1 over 7 open parentheses 1 half close parentheses to the power of 7................ close curly brackets
ln space 3 space equals space 2 open curly brackets 0.5 plus 0.042 plus 0.006 plus 0.001 plus...... close curly brackets space space equals space 1.098 end style
 
similarly if we put x = 4 in the above log series, we get ln 4
Answered by Thiyagarajan K | 29 Nov, 2018, 05:58: AM

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