\begin{tabular}{lll|l|l|l|l|l} Converges at: & \( T=326.70 K \) & \( \gamma_{1}=1.3629 \) & \( \gamma_{2}=1.2523 \) & \( x_{1}=0.4602 \) & \( x_{2}=0.5398 \) \\ \cline { 1 - 4 } & & \end{tabular} (i) \( \times \) Collision Scheme NTC The Direct Simulation Monte Carlo (DSMC) collisional scheme SBT. kirilshterev.com OPEN
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The serien $\sum_{n=0}(2 x)^{n}$ converges if $(a)-1 \leq x \leq 1$ $\begin{array}{lll}\text { (b) }-\frac{1}{2}<x<\frac{1}{2} & \text { (c) }-2<x<2 & \text { (d) }-\frac{1}{2} \leq x \leq \frac{1}{2} \text {. }\end{array}$
Define $J=\int_{0}^{\infty} \frac{d x}{x^{1 / 2}(x+1)}$ as the sum of the two improper integrals $$\int_{0}^{1} \frac{d x}{x^{1 / 2}(x+1)}+\int_{1}^{\infty} \frac{d x}{x^{1 / 2}(x+1)}$$ Use the Comparison Test to show that J converges.
TECHNIQUES OF INTEGRATION
Improper Integrals
Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{\infty} \frac{1}{4+x^{2}} d x $$
Techniques of Integration
Improper Integrals—Unbounded Intervals
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