812,000 gallons of water in an olympic training center's pool. express this number in scientific notation.
2 answers:
Answer: In scientific notation = 8.12\times10^5 gallons of water in an olympic training center's pool.
Step-by-step explanation:
Given: 812,000 gallons of water in an olympic training center's pool.
In scientific notation ,
Scientific notation is a way to write a very large or a very small number in the product of a decimal form of number [generally between 1 and 10]and powers of ten.
Therefore,In scientific notation = 8.12\times10^5 gallons of water in an olympic training center's pool.
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The power series is
g(x) = -2x - (2x)2/2 - (2x)3/3 -(2x)4/4 -.......-(2x)n/n - .....
To deduce the power series of g(x) from the power series for f(x) and identify its radius of convergence
The power series for f(x) is just the geometric series derived from 1/1-y ,setting y=2x.
Its radius of convergence is 0.5
Let,
f(x)= 1/1-2x = 1+ (2x) + (2x)2 + .........+(2x)n......+....
The power series expansion (geometric series),
valid for I2xI < 1 , IxI < 0.5
so, radius of convergence = 0.5
The power series for g(x) is found by integrating term by term the power series of f(x) (upto a constant). The radius of converngence of g(d) is the same as that of f(x) (from general theory) =0.5
Now, g(x) = ln(1-2x)
= -2
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g(x) = -2x - (2x)2/2 - (2x)3/3 -(2x)4/4 -.......-(2x)n/n - .....
is the power series expansion for g(x).
radius of convergence =0.5
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