\(1).\) A necessary condition for convergence of an infinite series \(\sum a_n\) is that $$\lim_{n \to \infty}a_n = 0.$$
\(2).\) A series \(\sum a_n\) converges iff for each \(\epsilon > 0\), there exists a positive integer \(m\) such that $$|a_{n+1} + u_{n+2} + \cdots +u_{n+k}| < \epsilon, \; \forall n \geq m \; and \; k \geq 1$$
\(3).\) If \(\sum u_n = u\), then \(\sum bu_n = bu\), independednt of \(n\)
\(4).\) Convergetn series may be added or subtracted term by term, if \(\sum u_n = u\) and \(\sum v_n = v\) then \(\sum w_n = u \pm v\), where \(w_n = u_n \pm v_n\), for all \(n\).
\(i).\) if any two of three series are convergent, the third is alos convergent
\(ii).\) if one of the series is divergent and another convergent then the third is necessarily divergent, but
\(iii).\) if two of the series are divergent, no conclusion can be drawn about the behaviour of the third, which may converge or diverge.
\(5).\) If a series \(\sum a_n\) converges to the sum \(u\) then so does any series obtained from \(\sum a_n\) by grouping the terms in brackets without altering the order of the terms.
\(6)\). A positive term series convergent iff the sequence of its partial sums is bounded above.
\(7).\) \(\textbf{Pringsheim's theorem}\) : If a series \(\sum a_n\) of poistive monotonic decreasing terms converges then not only \(a_n \rightarrow 0\) but also \(nu_n \rightarrow 0\) as \( n \rightarrow \infty\).
\(8).\) A positive term series \(\displaystyle \sum \frac{1}{n^p} \) is convergent iff \(p>1\).
\(9).\) Let \( \sum a_n \) and \( \sum b_n \) be series with positive terms, i.e., \( a_n \geq 0 \) and \( b_n \geq 0 \) for all \( n \in \mathbb{N} \). Suppose that there exists a constant \( C > 0 \) such that
\[
a_n \leq C b_n \quad \text{for all sufficiently large } n.
\]
Then:
\(i).\) If \( \sum b_n \) converges, then \( \sum a_n \) also converges.
\(ii).\) If \( \sum a_n \) diverges, then \( \sum b_n \) also diverges.
\(10).\) Let \( \sum a_n \) and \( \sum b_n \) be series with positive terms, i.e., \( a_n \geq 0 \) and \( b_n \geq 0 \) for all \( n \in \mathbb{N} \) such that \(\displaystyle \lim_{n \to \infty} \left(\frac{a_n}{b_n}\right)=l\), where \(l\) is a non zero finite number, then the two series converges or diverges togather.
\(11).\) \(\textbf{Limit Comparison Test}\) :
Let \( \sum a_n \) and \( \sum b_n \) be series with positive terms, i.e., \( a_n > 0 \) and \( b_n > 0 \) for all \( n \in \mathbb{N} \) and there exists a positive integer \(m\) such that
\[ \frac{a_n}{a_{n+1}} \geq \frac{b_n}{b_{n+1}}, \forall n \geq m
\]
Then
\(i).\) \(\sum a_n\) is convergent, if \(\sum b_n\) is convergent.
\(ii).\) \(\sum b_n\) is divergent, if \(\sum a_n\) is divergent.
\(12).\) \(\textbf{Cauchy's Root Test}\) : Let \( \sum a_n \) be a series with positive terms, Define
\[
L = \lim_{n \to \infty} \sqrt[n]{a_n}.
\]
Then:
\(i).\) If \( L < 1 \), then the series \( \sum a_n \) converges.
\(ii).\) If \( L > 1 \), then the series \( \sum a_n \) diverges.
\(iii).\) If \( L = 1 \), the test is inconclusive.
\(13).\) \(\textbf{Theorem (D'Alembert's Ratio Test):} \) Let \( \sum a_n \) be a series with positive terms. Suppose the limit
\[
L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}
\]
exists.
Then
\(i).\) If \( L < 1 \), then the series \( \sum a_n \) converges.
\(ii).\) If \( L > 1 \), then the series \( \sum a_n \) diverges.
\(iii).\) If \( L = 1 \), the test is inconclusive.
\(14).\ \textbf{Raabe's Test} :\)
Let \( \sum a_n \) be a series with positive terms, i.e., \( a_n > 0 \) for all \( n \in \mathbb{N} \). Define
\[
\lim_{n \to \infty} n\left( \frac{a_n}{a_{n+1}} - 1 \right) = l.
\]
Then
\(i).\) If \(l >1\) then the series \( \sum a_n \) is convergent.
\(ii).\) If \( l < 1 \),then series \( \sum a_n \) is divergent.
\(iii).\) If \( l = 1 \), the test is inconclusive.
\(15).\ \textbf{Logarithmic Test} :\)
Let \( \sum a_n \) be a series with positive terms, i.e., \( a_n > 0 \) for all \( n \in \mathbb{N} \), and suppose the limit
\[
L = \lim_{n \to \infty} n \log\left( \frac{a_n}{a_{n+1}} \right)
\]
exists.
\(i).\) If \( L > 1 \), then the series \( \sum a_n \) is convergent.
\(ii).\) If \( L < 1 \), then the series \( \sum a_n \) is divergent.
\(iii).\) If \( L = 1 \), the test is inconclusive.
\(16).\ \textbf{Integral Test} :\)
Let \( f(x) \) be a non negative, continuous, monotonic decreasing function and let \( a_n = f(n) \), for all positive integral values of \(n\), then the series \( \sum a_n \) and the improper integral \( \displaystyle\int_1^\infty f(x) \, dx \) either both converge or both diverge.
\(i).\) If \( \displaystyle \int_1^\infty f(x) \, dx \) converges, then \( \sum a_n \) converges.
\(ii).\) If \( \displaystyle\int_1^\infty f(x) \, dx \) diverges, then \( \sum a_n \) diverges.
\(17).\ \textbf{Gauss's Test} :\)
Let \( \{ a_n \} \) be a sequence of positive numbers. If
\[
\frac{a_n}{a_{n+1}} = \alpha + \frac{\beta}{n} + \frac{\gamma_n}{n^p}
\]
where \(\alpha >0, p>1,\) and \(\{\gamma_n\}\) is a bounded sequence, then
\(i).\) For \(\alpha \neq 1, \; \sum a_n \) converges if \(\alpha >1,\) and diverges if \(\alpha <1\), whatever \(\beta\) may be.
\(ii).\) For \(alpha =1\) , \(\sum a_n\) converges if \(\beta >1\), and diverges if \(\beta \leq 1\).
\(18).\) \(\textbf{Alternating series}\) : A series with alternating psoitive and negative terms, eg $$1-\frac{1}{2}+\frac{1}{4}-\cdots$$
\(19).\ \textbf{Alternating Series Test (Leibniz's Test)} :\)
Let \( \sum (-1)^{n+1} a_n \) be an alternating series, where \( a_n > 0 \). If the sequence \( \{a_n\} \) satisfies:
\(i).\) \( a_{n+1} \leq a_n \) for all \( n \in \mathbb{N} \) (i.e., the sequence is decreasing),
\(ii).\) \( \lim a_n = 0 \),
then the alternating series \( \sum (-1)^{n+1} a_n \) is convergent.
\(20).\) Absolute and Conditional Convergence :
Let \( \sum a_n \) be a series of real numbers.
Then
\(i).\) If the series \( \sum |a_n| \) converges, then the series \( \sum a_n \) also converges. In this case, the series is said to be absolutely convergent.
\(ii).\) If the series \( \sum a_n \) converges but the series \( \sum |a_n| \) diverges, then \( \sum a_n \) is said to be conditionally convergent.
\(21).\) Every absolutely convergent series is convergent.
\(22).\ \textbf{Abel's Test} :\) If \(a_n\) is a positive monotonic decreasing function and if \(\sum b_n\) is a convergent series, then the series \(\sum a_n b_n\) is also convergent.
\(23).\) A convergent series \(\sum a_n\) ( which may or may not be absolutely convergent) will be convergent if its terms are each multiplied by a factor \(b_n\), provided that sequence \(\{b_n\}\) is bounded and monotonic.
\(24).\ \textbf{Dirichlet's Test} :\)
Let \( \sum a_n b_n \) be a series where:
\(i).\) The sequence of partial sums \( A_n = \sum_{k=1}^n a_k \) is bounded,
\(ii).\) The sequence \( \{b_n\} \) is positive, monotonic decreasing,
\(iii).\) \( \lim b_n = 0 \),
then the series \( \sum a_n b_n \) is convergent.
\(25).\ \textbf{Rearrangement of Terms in a Series} :\)
Let \( \sum a_n \) be a given series. A series \( \sum a_{n_k} \) is called a rearrangement of \( \sum a_n \) if \( \{n_k\} \) is a bijective mapping (i.e., a one-to-one and onto permutation) of \( \mathbb{N} \), meaning every term of the original series appears exactly once in the rearranged series.
In other words, the rearrangement does not omit or repeat any terms; it only changes the order of summation.
\(26).\) A series obtained from an absolutily convergent series by a rearrangmet of terms converges absolutly and has the same sum as the original series.
\(27). \; \textbf{Riemann's theorem}\) : By an appropriate rearrangement of terms, a conditionally convergent series \(\sum a_n\) can be made
\(i)\). converges to any number \(l\).
\(ii)\). diverges to \(\infty\).
\(iii)\). diverges to \(-\infty\).
\(iv)\). oscillate finitely.
\(v)\). oscillate infinitely.
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