Chapter9: Sequences, Probability And Counting Theory
Section9.4: Series And Their Notations
Problem 10TI: Determine whether the sum of the infinite series is defined. 24+(12)+6+(3)+
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Question
Transcribed Image Text:The nth term of the series sin-
n=1
has zero limit
O has a limit of (+1)
O has no limit
O has a limit of (-1)
Expert Solution
Data analysis
Note : The sin(angle) will be always positive if the angle lies in first and second quadrant.
The sin(angle) will be always negative if the angle lies in third and fourth quadrant.
The above plot will explain this question perfectly.
Explanation
The nth term of the given series is,
nth term = sin (nπ/2) , where n = 1 to infinite
If n=1, 1st term = sin(π/2) = +1
If n=2, 2nd term = sin(2π/2) = 0
If n=3, 3rd term = sin(3π/2) = -1
If n=4, 4th term = sin(4π/2) = 0
If n=5, 5th term = sin(5π/2) = +1
If n=6, 6th term = sin(6π/2) = 0
If n=7, 7th term = sin(7π/2) = -1
.... and goes on till infinity.
Hence the value will oscillate between (+1, 0,-1) and never limited to a single value as 'n' goes on increasing.
That is,
- If 'n' is even number, then output value will be 0.
- If 'n' is (even number-1) , then output will be +1.
- If 'n' is (even number+1), then output will be -1.
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