Calculați a+b, unde:​

[tex]a = 1 + \frac{1}{2}x + \frac{1}{3}x^2 + \frac{1}{4}x^3 + \cdots + \frac{1}{31}x^{30}\\ b = \frac{1}{2}x + \frac{2}{3}x^2 + \frac{3}{4}x^3 + \cdots + \frac{30}{31}x^{30}\\ a+b = 1 + x(\frac{1}{2} + \frac{1}{2}) + x^2(\frac{1}{3} + \frac{2}{3}) + x^3(\frac{1}{4} + \frac{3}{4}) + \cdots + x^{30}(\frac{1}{31} + \frac{30}{31}) = 1 + x\cdot 1 + x^2\cdot 1 + x^3\cdot 1+ \cdots + x^{30} \cdot 1 = 1 + x + x^2 + x^3 + \cdots + x^{30} = \sum_{i=0}^{30} (x^i)[/tex]

Răspuns:

Explicație pas cu pas:

a=1+1/2x+1/3x²+.......+1/31x^30

b=1/x+2/3x²+......_30/31x^30

a+b=1+x²+x³+....+x^30  progresie geometrica cu ratia q=x; b1=1

S=(x^30-1)/(x-1)