1Q.2) 1/1.3 = 22-12 / 22.12 = 1/12 - 1/22
5/36 = 32-22 / 32.22 = 1/22 - 1/32
7/144 = 42-32 / 42.32 = 1/32 - 1/42
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UPTO n2-(n-1)2 / n2.(n-1)2 = 1/(n-1)2 - 1/n2
add them to get 1 - 1/n2.
when n→ ∞ , sum gets reduced to 1.
Ques1) Show that the sum of the series
1 / 1.3 + 2 / 1.3.5 + 3 / 1.3.5.7 + ........... upto n terms is 1/2.
Ques2) Show that the sum of n terms for series
3/4 + 5/36 + 7/144 + 9/400 + 11 / 900 + ......... is 1.
Ques3) If the angles of a triangle arein A.P, tangent of the samllest angle is 1, then find it's angles.
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3 Answers
1Q.3) Take three angles to be: a - d , a , a + d.sum of the angles is pi. so, a=pi / 3
tan ( a - d ) = 1, so,this angle is pi / 4. and the final angle is 750.
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341We are asked to find \sum_{k=1}^{\infty} \frac{k}{1\times 3 \times...\times 2k+1}
Notice that k = \frac{2k+1-1}{2}
So we are asked to find
\frac{1}{2} \sum_{k=1}^{\infty} \frac{2k+1-1}{1\times 3 \times...\times 2k+1} = \frac{1}{2} \sum_{k=1}^{\infty} \frac{1}{1 \times 3 \times ...\times (2k-1)} - \frac{1}{1 \times 3 \times ...\times (2k+1)}
which is a telescopic summation which sums to 1/2
