## 3 Convergence Analysis  
In this section, we prove the convergence analysis of the three I.F.s (11), (15) and (18).  
**Theorem 4**.: _Let a sufficiently smooth function \(f:D\subset\mathbb{R}\to\mathbb{R}\) has a simple root \(x^{*}\) in the open interval \(D\) which contains \(x_{0}\) as an initial approximation to \(x^{*}\). Then the \(5^{th}3 pVS\) I.F (11) is of local fifth-order convergence, the \(10^{th}4 pVS\) I.F (15) is of local 10 th-order convergence and the \(20^{th}5 pVS\) I.F (18) is of local 20 th-order convergence._  
**Proof.** Let \(e_{n}=x-x^{*}\).  
Using the [PERSON] series and the symbolic software such as Maple we have  
\[f(x)=f^{\prime}(x^{*})[e_{n}+c_{2}e_{n}^{2}+c_{3}e_{n}^{3}+c_{4}e_{n}^{4}+\ldots] \tag{19}\]  
and  
\[f^{\prime}(x)=f^{\prime}(x^{*})[1+2c_{2}e_{n}+3c_{3}e_{n}^{2}+4c_{4}e_{n}^{3}+\ldots]\]  
so that  
\[u(x)=e_{n}-c_{2}e_{n}^{2}+2(c_{2}^{2}-c_{3})e_{n}^{3}+(7c_{2}c_{3}-4c_{2}^{3}-3 c_{4})e_{n}^{4}+\ldots\quad.\]  
Now  
\[\psi_{2^{nd}NR}(x)=x^{*}+\,{c_{2}e_{n}}^{2}+\left(2\,c_{3}-2\,{c_{2}}^{2} \right)e_{n}{}^{3}+\left(3\,c_{4}-7\,c_{2}c_{3}+4\,{c_{2}}^{3}\right)e_{n}{}^ {4}+\ldots,\]  
so that  
\[\tau=\frac{f^{\prime}(\psi_{2^{nd}NR}(x))}{f^{\prime}(x)}=1-c_{2}e_{n}+\left( 6\,{c_{2}}^{2}-3\,c_{3}\right)e_{n}{}^{2}+\left(16\,c_{2}c_{3}-16\,{c_{2}}^{3 }-4\,c_{4}\right)e_{n}{}^{3}+O\left(e_{n}{}^{4}\right),\]  
\[H(\tau)=1+{c_{2}}^{2}e_{n}{}^{2}+\left(3\,c_{2}c_{3}-3\,{c_{2}}^{3}\right)e_{n }{}^{3}+O\left(e_{n}{}^{4}\right) \tag{20}\]  
and  
\[D(x)=f^{\prime}(x^{*})\Bigg{(}1+c_{2}\;e_{n}+\left({c_{2}}^{2}+c_{3}\right)e_ {n}{}^{2}+\left(3\,c_{2}c_{3}-2\,{c_{2}}^{3}+c_{4}\right)e_{n}{}^{3}+O\left(e _{n}{}^{4}\right)\Bigg{)}. \tag{21}\]  
Substituting Equations (19), (20) and (21) into Equation (11), we obtain, after simplifications,  
\[\psi_{5^{th}3 pVS}(x)-x^{*}=C_{5^{th}3 pVS}e_{n}{}^{5}+O\left(e_{n}{}^{6}\right),\quad C_{5^{th}3 pVS}=-\frac{1}{4}\,{c_{3}}^{2}-\frac{1}{2}\,{c_{3}}{c_{2}}^{ 2}+9\,{c_{2}}^{4}. \tag{22}\]  
Now, using Maple, we have  
\[G_{1}=\frac{1}{f^{\prime}(x^{*})}\left(1+C_{G_{1}}\;{e_{n}}^{2}+O\left(e_{n}{} ^{3}\right)\right),\quad C_{G_{1}}=-18\,{c_{2}}^{5}+{c_{3}}{c_{2}}^{3}+1/2\,{ c_{2}}{c_{3}}^{2}.\]  
Using [PERSON]'s theorem with \(p=q=5\), we have  
\[\psi_{10^{th}4 pVS}(x)-x^{*}\] \[=\left(-C_{G_{1}}\;C_{5^{th}3 pVS}-c_{2}\;C_{5^{th}3 pVS}^{2}\right) e_{n}{}^{10}+O\left(e_{n}{}^{11}\right)\] \[=C_{10^{th}4 pVS}\;e_{n}{}^{10}+O\left(e_{n}{}^{11}\right),\quad C _{10^{th}4 pVS}=-\frac{17}{4}\,{c_{3}}^{2}{c_{2}}^{5}-9\,{c_{3}}{c_{2}}^{7}+ \frac{1}{4}\,{c_{3}}^{3}{c_{2}}^{3}+81\,{c_{2}}^{9}+\frac{1}{16}\,{c_{2}}{c_{3 }}^{4}. \tag{23}\]  
Using Maple,  
\[G_{2} =\frac{1}{f^{\prime}(x^{*})}\left(1+C_{G_{2}}\;e_{n}{}^{10}+O \left(e_{n}{}^{11}\right)\right),\] \[C_{G_{2}} =-162\,{c_{2}}^{10}+18\,{c_{3}}{c_{2}}^{8}+\frac{17}{2}\,{c_{3}}^ {2}{c_{2}}^{6}-\frac{1}{2}\,{c_{3}}^{3}{c_{2}}^{4}-\frac{1}{8}\,{c_{3}}^{4}{c_{2 }}^{2}.\]Using [PERSON]'s theorem with \(p=q=10\), we have  
\[\begin{split}&\psi_{20^{th}5 pVS}(x)-x^{*}\\ &=(-C_{G_{2}}\,C_{10^{th}4 pVS}-c_{2}\,C_{10^{th}4 pVS}^{2})\,{e_{n}} ^{20}+O\left({e_{n}}^{21}\right)\\ &=C_{20^{th}5 pVS}\,{e_{n}}^{20}+O\left({e_{n}}^{21}\right),\\ & C_{20^{th}5 pVS}=6561{c_{2}}^{19}-1458{c_{3}}{c_{2}}^{17}-\frac{ 1215}{2}{c_{3}}^{2}{c_{2}}^{15}+117{c_{3}}^{3}{c_{2}}^{13}+\frac{379}{16}{c_{3 }}^{4}{c_{2}}^{11}-\frac{13}{4}{c_{3}}^{5}{c_{2}}^{9}\\ &\hskip 142.26378 pt-\frac{15}{32}{c_{3}}^{6}{c_{2}}^{7}+\frac{1}{ 32}{c_{3}}^{7}{c_{2}}^{5}+\frac{1}{256}\,{c_{3}}^{8}{c_{2}}^{3}\end{split} \tag{24}\]