Why the number3.456456... is irrationnal
1 answer:
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Answer:
Yes, it can be concluded that the mean hexane concentration is less in treated water than in unsaturated water
Step-by-step explanation:
The number of of specimen in the samples of untreated water, n₁ = 16
The sample mean, = 228.0
The sample standard deviation, s₁ = 4.3
The number of of specimen in the samples of treated water, n₂ = 20
The sample mean, = 224.6
The sample standard deviation, s₂ = 5.0
The level of significance = 0.10
The null hypothesis, H₀; ≥
The alternative hypothesis, Hₐ; <
The degrees of freedom = 16 - 1 = 15
The test statistic, = 1.341
Plugging in the values, we get;
Given that the t-value is large, the corresponding p-value is low, therefore, we fail to reject the null hypothesis and there is considerable statistical evidence to suggest that the mean hexane concentration is less in treated than in untreated water, therefore, we have; ≥
Answer:
The Taylor series is .
The radius of convergence is .
Step-by-step explanation:
<em>The Taylor expansion.</em>
Recall that as we want the Taylor series centered at its expression is given in powers of
. With this in mind we need to do some transformations with the goal to obtain the asked Taylor series from the Taylor expansion of
.
Then,
Now, in order to make a more compact notation write . Thus, the above expression becomes
Notice that, if x is very close from 3, then y is very close from 0. Then, we can use the Taylor expansion of the logarithm. Hence,
Now, substitute in the previous equality. Thus,
<em>Radius of convergence.</em>
We find the radius of convergence with the Cauchy-Hadamard formula:
Where stands for the coefficients of the Taylor series and
for the radius of convergence.
In this case the coefficients of the Taylor series are
and in consequence . Then,
Applying the properties of roots
Hence,
Recall that
So, as we get that
.