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(C0)/1. 3-(C1)/2. 3+(C2)/3. 3-(C3)/4. 3+...

`(C_0)/1. 3-(C_1)/2. 3+(C_2)/3. 3-(C_3)/4. 3+ +(-1)^n(C_n)/((n+1)*3)` 01+23+ (-1)1.3 2.3 3.3 4.3

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(C_0)/(1. 3)-(C_1)/(2. 3)+(C_2)/(3. 3)-(C_3)/(4. 3)+...... +(-1)^n(C_n)/((n+1)*3) is

(C_0)/(1. 3)-(C_1)/(2. 3)+(C_2)/(3. 3)-(C_3)/(4. 3)+...... +(-1)^n(C_n)/((n+1)*3) is

(C_(0))/(1)*3-(C_(1))/(2)*3+(C_(2))/(3)*3-(C_(3))/(4)*3++(-1)^(n)(C_(n))/((n+1)*3)01+23+(-1)1.32.33.34.3

(C_1)/(C_0)+2. (C_2)/(C_1)+3. (C_3)/(C_2)+….+n.(C_n)/(C_(n-1))=

2.(C_2)/(C_1)+3.(C_3)/(C_2)+…...+n. (C_n)/(C_(n-1))=

C_1/C_0 + (2C_2)/C_1 + (3C_3)/C_2 + .... (nC_n)/C_(n-1) is

k. C_0 + k^2 . (C_1)/(2)+k^3. (C_2)/(3)+…..+ k^(n+1). (C_n)/(n+1)=

(C_(0))/(1.2)+(C_(1))/(2.3)+(C_(2))/(3.4)+......*(C_(n))/((n+1)(n+2))=

2C_0 + 2^2 (C_1)/(2) + 2^3 (C_2)/(3) + ………. + 2^(n+1) (C_n)/(n+1) = (3^(n+1) - 1)/(n+1)