Find the sum of the series
-4 + 1 -1/4 + 1/16 - 1/64 + 1/256 + ... .
A GS, with
a = -4
r = -1/4
sum(∞) = a/(1-r)
= -4/(1 -(-1/4) )
= -4/(5/4)
= -16/5 or -3.2
Ah, series! They're like a never-ending party, except with numbers instead of people dancing! Now, let's crunch some numbers.
This series is like a seesaw. Whenever you add the numbers, they either bring the sum up or down. It's quite the balancing act!
We'll start by noticing a pattern. Each term is alternating between positive and negative, and the denominators are getting bigger and bigger.
The sum of this series can be found by using the formula for the sum of an infinite geometric series:
S = a / (1 - r)
Here, "a" is the first term (-4), and "r" is the common ratio (-1/4).
Plugging in the values, we get:
S = -4 / (1 - (-1/4))
S = -4 / (1 + 1/4)
S = -4 / (5/4)
S = -4 * (4/5)
S = -16/5
So, the sum of this series is -16/5. That's like a clown juggling numbers while riding a unicycle! Quite the wild ride, isn't it?
To find the sum of the series -4 + 1 -1/4 + 1/16 - 1/64 + 1/256 + ..., we can use the formula for the sum of an infinite geometric series.
The formula for the sum of an infinite geometric series is given by:
S = a / (1 - r),
where:
- S is the sum of the series,
- a is the first term of the series, and
- r is the common ratio between terms.
In this case, the first term (a) is -4 and the common ratio (r) is -1/4.
Plugging these values into the formula, we get:
S = -4 / (1 - (-1/4)).
Simplifying further:
S = -4 / (1 + 1/4).
S = -4 / (5/4).
To divide by a fraction, we can multiply by its reciprocal:
S = -4 * (4/5).
S = -16/5.
Therefore, the sum of the series -4 + 1 -1/4 + 1/16 - 1/64 + 1/256 + ... is -16/5.
To find the sum of the given series, we notice that it is a geometric series with a common ratio of -1/4.
A geometric series has the form: a + ar + ar^2 + ar^3 + ...
In this case, the first term (a) is -4 and the common ratio (r) is -1/4. Therefore, we can rewrite the series as:
-4 + (-4)(-1/4) + (-4)(-1/4)^2 + (-4)(-1/4)^3 + ...
To find the sum of the series, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
In this case, plugging in the values, we have:
S = -4 / (1 - (-1/4))
Simplifying the denominator, we get:
S = -4 / (1 + 1/4)
Combining the fractions on the denominator, we have:
S = -4 / (5/4)
To divide by a fraction, we can multiply by its reciprocal:
S = -4 * (4/5)
Multiplying the numerator and denominator, we get:
S = -16/5
Therefore, the sum of the given series is -16/5.