There were 160,000 deaths from cardiovascular causes in the UK in 2016. If the number of deaths decreases by 5% every year, year after year, indefinitely, what is the total number of people that would have succumbed to cardiovascular causes?
This is an example of a convergent geometric series.
In general, the sum of a geometric series is given by S= a (1-r^n)/1-r, where S is the sum of the first n numbers, a is the first number in the sequence, and r is the common ratio of the sequence (the factor by which each successive number increases or decreases).
Now, can you predict what would be the sum of the series if n was infinite? Would the sum S be finite or infinite?
The answer is that it depends on the value of r, the common ratio. If r is between -1 & 1, the sum S would be a finite (convergent series), while if r is <-1 or >1, the sum would be infinite (divergent series).
If r is a fraction, the limit of the equation S= a (1-r^n)/1-r, as n approaches infinity, is S = a/1-r.
In the above example, r=0.95, as the death rate falls by 5% every year, the total number deaths in infinite number of years would be 160,000/0.05=32,00,000.
The sum of this geometric series is finite, hence it's a convergent series.
This is an example of a convergent geometric series.
In general, the sum of a geometric series is given by S= a (1-r^n)/1-r, where S is the sum of the first n numbers, a is the first number in the sequence, and r is the common ratio of the sequence (the factor by which each successive number increases or decreases).
Now, can you predict what would be the sum of the series if n was infinite? Would the sum S be finite or infinite?
The answer is that it depends on the value of r, the common ratio. If r is between -1 & 1, the sum S would be a finite (convergent series), while if r is <-1 or >1, the sum would be infinite (divergent series).
If r is a fraction, the limit of the equation S= a (1-r^n)/1-r, as n approaches infinity, is S = a/1-r.
In the above example, r=0.95, as the death rate falls by 5% every year, the total number deaths in infinite number of years would be 160,000/0.05=32,00,000.
The sum of this geometric series is finite, hence it's a convergent series.
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