Zara Patel

I'm studying abstract algebra and I understand the group axioms: 1. **Closure:** $a \cdot b \in G$ for all $a, b \in G$ 2. **Associativity:** $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ 3. **Identity:** There exists $e \in G$ such that $e \cdot a = a \cdot e = a$ 4. **Inverses:** For each $a$, there exists $a^{-1}$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$

I'm studying number theory and I need to solve linear congruences like: $$17x \equiv 3 \pmod{29}$$

I understand how to compute Laplace transforms using the formula: $$\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st} f(t) \, dt$$

I'm studying sequences and series in calculus. There are so many tests for convergence — the divergence test, ratio test, root test, integral test, comparison test, alternating series test — and I'm overwhelmed.

Could someone provide a decision tree or systematic approach for determining which test to apply to a given series?

I am struggling with the $\varepsilon-\delta$ definition of a limit in my Real Analysis class: $$\lim_{x \to a} f(x) = L \iff \forall \varepsilon > 0, \exists \delta > 0 \text{ such that } 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon$$

I'm revising A-Level integration and I keep getting stuck on integrals of the form: $$\int \frac{1}{x^2 + a^2} \, dx$$