Ex. 1, Find whether the series whose nth term is (n+1)x^n n^2 convergent or divergent.
Here u_n u_n-1 = {(n+1)x^n n^2} {(n-1)^2 nx^n-1} = {(n+1) (n-1)^2 x }{n^3}; Lim u_n u_n-1 = x;
hence if x< 1 the series is convergent ; if x > 1 the series is divergent.
If x = 1, then Lim u_n u_n-1 = 1, and a further test is required.
Ex. 2. Is the series 1^2 + 2^2x + 3^2x^2 + 4^2x^3 + ... convergent or divergent ?
Here Lim u_n u_n-1 = Lim = x. Hence if x<1 the series is convergent ; if x > 1 the series is divergent. If x = 1 the series becomes 1^2 + 2^2 + 3^2 + 4^2+ ... , and is obviously divergent.
Ex. 3. In the series a + (a+d)r + (a+2d)r^2 + ... + (a + n - 1 . d)r^{n-1} + ..., Lim u_n u_n-1 = Lim = r; thus if r < 1 the series is convergent, and the sum is finite.
477. Fifth Test. If there are two infinite series in each of which all the terms are positive, and if the ratio of the corresponding terms in the two series is always finite, the two series are both convergent, or both divergent.
Let the two infinite series be denoted by
u_1 + u_2 + u_3 + u_4 + ... and v_1 + v_2 + v_3 + v_4 + ...
The value of the fraction
{u_1 + u_2 + u_3 + u_4 + ... + u_n} {v_1 + v_2 + v_3 + v_4 + ... + v_n}
