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Card 9 / 9:
remainder estimate
for a series ∑ n = 1 ∞ a n with positive terms a n and a continuous, decreasing function f such that f ( n ) = a n for all positive integers n , the remainder R N = ∑ n = 1 ∞ a n − ∑ n = 1 N a n satisfies the following estimate: ∫ N + 1 ∞ f ( x ) d x < R N < ∫ N ∞ f ( x ) d x
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