The value of the expression ∑ r = 0 ∞ 1 2 r ∑ k = 0 ∞ 1 3 k ∑ P = 0 ∞ 2 3 P is

Question

The value of the expression ∑ r = 0 ∞ 1 2 r ∑ k = 0 ∞ 1 3 k ∑ P = 0 ∞ 2 3 P is, 1, 2, 1 2, 4 3

MARKS App · Accepted Answer

∑ r = 0 ∞ 1 2 r = 1 + 1 2 + 1 2 2 . . . . ∞ = a 1 - r = 1 1 - 1 2 = 2 (Applying formula of the sum of infinite G.P.) Similarly, ∑ K = 0 ∞ 1 3 K = 1 1 - 1 3 = 3 2 & ∑ P = 0 ∞ 2 3 P = 1 1 - 2 3 = 3 Hence, the expression simplifies to 2 × 3 2 3 = 1

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