y= ¡Ìsin x+¡Ìsin x+sin x+.............infinity
To find the value of the infinite sum y = √sin(x) + √sin(x) + sin(x) + ..., we can use a concept known as an infinite geometric series.
An infinite geometric series is a sum of terms where each term is obtained by multiplying the previous term by a constant factor. In this case, we need to determine if the sum is convergent or divergent.
Let's denote the common ratio as r, which is the value by which each term is multiplied to obtain the next term. In our case, the terms are √sin(x), √sin(x), sin(x), ... and the common ratio is √sin(x)/sin(x).
Taking the limit as the number of terms approaches infinity, we have:
lim(n→∞) (√sin(x)/sin(x))^n.
To determine the value of this limit, we need to distinguish the cases where sin(x) is equal to zero and where sin(x) is not zero.
Case 1: sin(x) = 0
If sin(x) = 0, then all the terms in the sum are equal to zero. In this case, the sum y is equal to zero.
Case 2: sin(x) ≠ 0
If sin(x) is not zero, we can simplify the limit as follows:
lim(n→∞) (√sin(x)/sin(x))^n = (1/√sin(x))^n = (sin(x))^(-1/2 * n).
For this limit to converge, the absolute value of sin(x) must be strictly less than 1. In other words, |sin(x)| < 1. If this condition is not met, the sum will diverge.
If |sin(x)| < 1, then the limit will approach zero as n approaches infinity. Hence, the sum y will converge to zero in this case.
To summarize, the infinite sum y = √sin(x) + √sin(x) + sin(x) + ... is convergent and its sum is zero when |sin(x)| < 1. Otherwise, the sum diverges.