- The first equation is incorrect. The limit $\lim_{x \to \infty} \frac{\pi(x)}{x / \ln(x)} = 1$ is the statement of the Prime Number Theorem, where $\pi(x)$ counts the number of primes less than or equal to $x$. The symbol in the image is $\Pi(x)$, not $\pi(x)$. $\Pi(x)$ typically denotes the prime-counting function weighted by powers of primes (i.e., $\Pi(x) = \sum_{p^k \le x} \frac{1}{k}$), and its asymptotic behavior is also $\sim \frac{x}{\ln x}$, so the limit would still be 1 if interpreted as $\Pi(x)$. However, given the standard notation, this is likely a typo, and it should be $\pi(x)$. Assuming the intended meaning is the Prime Number Theorem, the limit is indeed 1.
- The second equation is correct. It states that the sum of binomial coefficients $\sum_{k=0}^{n} \binom{n}{k} = 2^n$. This follows from the Binomial Theorem: $(1+1)^n = \sum_{k=0}^{n} \binom{n}{k} 1^k 1^{n-k} = \sum_{k=0}^{n} \binom{n}{k} = 2^n$.
- The third equation is incorrect. The contour integral $\oint_C \frac{f'(z)}{f(z)} dz = 2\pi i (N - P)$ is the Argument Principle, which holds when $f$ is meromorphic inside and on a simple closed contour $C$, with no zeros or poles on $C$. Here, $N$ is the number of zeros and $P$ is the number of poles of $f$ inside $C$, each counted with multiplicity. The equation as written is missing the crucial context that $f$ must be meromorphic and $C$ must enclose the zeros and poles without passing through them. Without these conditions, the equation is not generally true.
Parent Tip: Review the logic above to help your child master the concept of equation math.
