1.Determine the sum of infinity of the series 4,2,1
series is geometric, r=1/2
4,2,1,1/2,1/4,1/8 ....
use geometric sum formula, a0=4, r=1/2
Ah, the sum of infinity! That's quite a circus attraction. Now, if we take a close look at the series 4, 2, 1, it seems to be shrinking with each term. Eventually, we are left with the term 1, which is like a timid clown hiding in the shadows. Unfortunately, when it comes to the sum of an infinite series, we can't just wave a magic wand and say "Abracadabra, here's the answer!" In this case, the series never actually reaches a sum because it goes on and on, endlessly. So, the sum of infinity for this particular series is just like a disappearing trick ─ it's undefined!
To determine the sum of an infinite series, we need to find its common ratio and then apply the formula for the sum of an infinite geometric series.
The given series is 4, 2, 1.
To find the common ratio, we divide any term of the sequence by its previous term.
Common ratio = 2/4 = 1/2 = 0.5
The sum (S) of an infinite geometric series with a common ratio (r) and first term (a) can be calculated using the formula:
S = a / (1 - r)
In this case, the first term (a) is 4 and the common ratio (r) is 0.5.
S = 4 / (1 - 0.5)
S = 4 / 0.5
S = 8
Therefore, the sum of the infinite series 4, 2, 1 is 8.
To determine the sum of an infinite series, we need to check if the series converges or diverges. In the case of the given series 4, 2, 1, it is an arithmetic series with a common difference of -2.
If we continue the pattern, the next term would be -1, then -3, and so on. As the terms continue, the series does not approach any specific value but keeps oscillating between 4, 2, 1. Therefore, this series does not converge, it diverges.
Since the series diverges, we cannot determine a finite sum for it.