For example, if you know that a1 <= b1 and a2 <= b2 then you can derive that a1+a2 <= b1+b2. The former implies the latter by simply adding the inequalities. What is the difference between bounding each term and bounding the sum of the series? But the statement of the problem presumes a_(k+1)/a_k <= r < 1 which for non-negative numbers implies a_(k+1) < a_k so the sequence of terms is strictly decreasing. The summation symbol does not indicate anything about the terms of the series. If the answer to Q1 is that it holds only for a decreasing arithmetic series then how does the summation symbol imply that ? Suppose that a_(k+1)/a_k = 0, otherwise the conclusion does not follow. It follows directly from this assumption that a_(k+1) < (a_k)r for every k, which means that a_k < a_0(r^k). This is because of the assumption that the ratio of successive terms of a_k is always less than or equal to r.That's what is happening here: the first inequality claims that the sum of the series on the left is bounded by the sum of the series on the right because each term (up to n) on the left is less or equal to than each corresponding term on the right (and after n, adding positive values retains the inequality). If every term is smaller than the corresponding term of another series, then the sum of the former will be smaller than the sum of the latter. n-th term of an AGP is denoted by: t n a + (n 1) d (b r n-1) Method 1: (Brute Force) The idea is to find each term of the AGP and find the sum. For example Counting Expected Number of Trials until Success. Bounding each term is a way of bounding the series. Arithmeticogeometric sequences arise in various applications, such as the computation of expected values in probability theory. So for, the above formula, how did they get (n + 1) ( n + 1) a for the geometric progression when r 1 r 1.What is the formula for the sequence Each term is obtained by adding 2 to the previous term. The condition on the ratio of successive terms in the sequence being between 0 and 1 does mean that the sequence is decreasing. Given a sequence finding a rule for generating the sequence is not always straightforward Example: Assume the sequence: 1,3,5,7,9.In other words, you can only bound with a decreasing geometric series (using this method) if there exists a constant r between 0 and 1 such that the ratio between successive terms is smaller than the constant r. The bounds on r are a precondition on using this method of bounding with a decreasing geometric series.
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