[from n=4 to infinity] E (2/3)3n-1 (-4/7)2n+3
*that E just means that sum symbol.
Let\'s say the sum of the series is S. Hence S will be as shown on the white board.
Please note the second term of the series. It\'s raised to the power of 2n + 3 which will always be an odd number. Hence, the second term that contains a negative sign inside the exponent, will always result into a negative term.
Hence, the series S will now appear as shown on the whiteboard.
S1 will have the following terms: (2/3)3x4-1 , (2/3)3x5-1 ,(2/3)3x6-1 and so on which can be simplified as: (2/3)11 , (2/3)14 ,(2/3)17 , and so on......
Thus, S1 is a geometric progression with first term, A = (2/3)11 and common Ratio, R = [(2/3)14] / [(2/3)11] = (2/3)3
Sum of an infinite geometric series with first term as A and common ratio R is given by A / (1 - R)
Hence, S1 = A / (1 - R) = (2/3)11 /[ 1- (2/3)3 ]
S2 will have the following terms: (4/7)2x4+3 , (4/7)2x5+3 ,(4/7)2x6+3 and so on which can be simplified as: (4/7)11 , (4/7)13 ,(4/7)15 , and so on......
Thus, S2 is a geometric progression with first term, A = (4/7)11 and common Ratio, R = [(4/7)13] / {(4/7)11] = (4/7)2
Sum of an infinite geometric series with first term as A and common ratio R is given by A / (1 - R)
Hence, S2 = A / (1 - R) = (4/7)11 /[ 1- (4/7)2 ]
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