Why did profits for growing crops increase in jamestown?
1) new agricultural products were grown in the colony.
2) growers increased the number of indentured servants.
3) laws were changed establishing lifelong slavery.

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Ответ:

tylerwoodson

04.04.2022

Mathematics

Step-by-step explanation:

Given a series $\sum_{k=1}^\infty f(z)$ , the Weierstrass M-test tell us that if we find a sequence of positive numbers M_n such that $|f(z)|\leq M_n$ in a certain domain D, and the series $\sum_{n=1}^\infty M_k$ converges, then the series $\sum_{k=1}^\infty f_k(z)$ converges uniformly in the domain D.

So, our objective is to find the so called sequence M_k . The main idea is to bound the sequence of functions $\frac{z^k}{z^k + 1}$ .

Now, notice that the values of z are always positive, so z^k is always positive, so $z^k+1\geq 1$ for all values of z in $\overline{D}$ . Then,

$\Big| \frac{z^k}{z^k + 1}\Big| \leq z^k,$

because if we make the values of the denominator smaller, the whole fraction becomes larger.

Moreover, as z is in the interval [0,r], we have that $z\leq r$ and as consequence $|z^k|\leq r^k$ . With this in addition to the previous bound we obtain

$\Big| \frac{z^k}{z^k + 1}\Big| \leq |z^k|\leq r^k.$

With this, our sequence is M_k = r^k and the corresponding series is $\sum_{k=1}^\infty r^k$ , which is a geometric series with ratio less than 1, hence it is convergent.

Then, as consequence of Weierstrass M-test we have the uniform convergence of the series in the given domain.

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Why did profits for growing crops increase in jamestown? 1) new agricultural products were grown in the colony. 2) growers increased the number of indentured servants. 3) laws were changed establishing lifelong slavery.