Let f(x)= ∑n=1∞ (-1)^n (x/4)^n. Find the series and interval of convergence for the integral from 0 to x of f(t)dt

# Tag: series

## google apps script – How do I code 3 series dependent drop down on multiple sheets?

I have a code to run a 3 tier dependent drop down list – it currently works for one sheet in my workbook named “BlankNode” but I need it to work for ~20 different sheets. I tried adding a script for each sheet but only the last one I edit will work. I think I need a loop function for a range of sheets but that doesn’t seem to be working. Am I able to get the below code to run on multiple sheets?

```
var mainWsName = "BlankNode";
var optionsWsName = "options"
var firstLevelColumn = 2;
var secondLevelColumn = 3;
var thirdLevelColumn = 4;
var ws = SpreadsheetApp.getActiveSpreadsheet().getSheetByName(mainWsName);
var wsOptions = SpreadsheetApp.getActiveSpreadsheet().getSheetByName(optionsWsName);
var options = wsOptions.getRange(2, 1, wsOptions.getLastRow()-1,3).getValues();
function onEdit(activeCell){
var activeCell = ws.getActiveCell();
var val = activeCell.getValue();
var r = activeCell.getRow()
var c = activeCell.getColumn()
var wsName = activeCell.getSheet().getName();
if(wsName === mainWsName && c === firstLevelColumn && r > 1){
applyFirstLevelValidation(val,r);
}else if(wsName === mainWsName && c === secondLevelColumn && r > 1){
applySecondLevelValidation(val,r);
}
}// end onEdit
function applyFirstLevelValidation(val,r){
if(val == ""){
ws.getRange(r,secondLevelColumn).clearContent();
ws.getRange(r,secondLevelColumn).clearDataValidations();
ws.getRange(r,thirdLevelColumn).clearContent();
ws.getRange(r,thirdLevelColumn).clearDataValidations();
} else {
ws.getRange(r,secondLevelColumn).clearContent();
ws.getRange(r,thirdLevelColumn).clearContent();
ws.getRange(r,secondLevelColumn).clearDataValidations();
ws.getRange(r,thirdLevelColumn).clearDataValidations();
var filteredOptions = options.filter(function(o){ return o(0) === val });
var listToApply = filteredOptions.map(function(o){ return o(1) });
var cell = ws.getRange(r,secondLevelColumn);
applyValidationToCell(listToApply,cell);
}
}
function applySecondLevelValidation(val,r){
if(val == ""){
ws.getRange(r,thirdLevelColumn).clearContent();
ws.getRange(r,thirdLevelColumn).clearDataValidations();
} else {
ws.getRange(r,thirdLevelColumn).clearContent();
var firstLevelColValue = ws.getRange(r, firstLevelColumn).getValue();
var filteredOptions = options.filter(function(o){ return o(0) === firstLevelColValue && o(1) === val });
var listToApply = filteredOptions.map(function(o){ return o(2) });
var cell = ws.getRange(r,thirdLevelColumn);
applyValidationToCell(listToApply,cell);
}
}
function applyValidationToCell(list,cell){
var rule = SpreadsheetApp
.newDataValidation()
.requireValueInList(list)
.setAllowInvalid(false)
.build();
cell.setDataValidation(rule);
}
```

## Do Program and Shutter priority modes work on a Nikon N90s with a Series E lens?

According to the Nikon Camera and Lens Compatibility Chart at Nikonians.org, **no**, you will not be able to use Program or Shutter-priority modes on your N90s with AI, AI-S, or E-series lenses.

Quoting a section from the chart:

Nikon Film SLR … AI,AI-S,E N90s/F90x … MF ^{1,2}

Notes

MFManual Focus1Only in A (Aperture Priority) or M (Manual) modes. P (Program) or S (Shutter priority) exposure modes will not function.2No 3D Matrix Exposure Metering.

## Do Priority and Shutter-release modes work on a Nikon N90s with a Series E lens?

According to the Nikon Camera and Lens Compatibility Chart at Nikonians.org, **no**, you will not be able to use Program or Shutter-priority modes on your N90s with AI, AI-S, or E-series lenses.

Quoting a section from the chart:

Nikon Film SLR … AI,AI-S,E N90s/F90x … MF ^{1,2}

Notes

MFManual Focus1Only in A (Aperture Priority) or M (Manual) modes. P (Program) or S (Shutter priority) exposure modes will not function.2No 3D Matrix Exposure Metering.

## real analysis – Interpretation of series solution

I have a function that is of the form

$f = a + b – cf$

I noticed that I can sove this in one of two ways. In the first way, the solution is

$f = frac{1}{1+c}(a+b)$

In the second way I can perform a recursion to get a series solution. There’s some intermediate steps to reach this realization, but in the end it has the form

$f = (a + b)*(1-c^2+c^4-c^6+ldots)$

I’ve checked and the series does in fact converge to the solution $1/(1+c)$ I presented prior. However, there is a slight difference. In the first solution, the domain for $c$ extends to infinity. In the second solution, the radius of convergence is 1, thus the domain for $c$ is bound. Here is the plot for a modest number of terms…

The convergence behavior is clear and everything to this point I’m good with. My question is centered around how to interpret the solution differences, specifically their domains. As shown in the figure, the domain for the first solution is infinite. Let’s assume nothing about the recursive approach was known (which lead to the series solution). Is the first solution actually wrong?

I’m trying to understand if one authors approach is superior to the other. Is there a benefit to using one approach vs the other? Within the radius of convergence they’re both the same in the limit of infinite terms, but beyond the radius of convergence, one solution goes one way while the other goes another way. The physical application does have the potential to take on values larger than $c=1$ due to what $c$ represents. The graph tells me that if $c=2$, go with the first solution approach, but I want to make sure I’m not missing something fundamental (e.g. there is an implicit assumption that $c<1$ in the first approach).

## summation – How do I find the sum of the following series?

I have to find the sum of the series $sum_{n=1}^{infty}1/((n+2)(2n+5))$. I performed partial fraction decomposition and this is what I got:

$sum_{n=1}^{infty}(frac{1}{n+2}-frac{2}{2n+5})$. So that makes my series:

$sum_{n=1}^{infty}(frac{1}{3}-frac{2}{7})+(frac{1}{4}-frac{2}{9})…$

How do I proceed from here?

Thanks in advance!!

## Google Sheets Sigma notation to calculate series

I have to calculate a series but since the `Sigma`

icon represents ALL possible formulas in the UI, it’s very difficult to find a documentation on the actual "mathematical" summation symbol.

Let’s say I want to calculate something like the growth of an hypothetical animal that weighs 200g over 3 years.

My mathematical notation would be the following

But I don’t know how to represent that in Google Sheets. What I did find was `SERIESSUM`

however the documentation left me with more question than answers, I’m not sure if the "power sum" they mention is what I am looking for.

## functional programming – How to write a C program to print Pi basing on Madhava series

in this question, i have been tasked to write a program that prints pi basing on madava series, using pi() function and power() function.

The issue is that, when i use power(), i dont get the actual value of pi.

This is the source code i have written.

```
#include<stdio.h>
#include<math.h>
double pi();
int power(int x, int y);
int n;
int main()
{
double ret;
ret=pi();
printf("%.16f",ret);
}
double pi()
{
double j, result;
int x, k, n;
printf("Input value of n:");
scanf("%d",&n);
result=0;
for( k=0; k<n; k++ )
{
result+=(power((-1),(k))/ power((2*k+1.3),k)) ;
}
j=(sqrt(12))*result;
return j;
}
int power(int x, int y)
{
int res = 1;
for( y; y>0; y--)
{
res = res*x;
}
return res;
}
```

## malware – Sending meaningless Addresses requests to a series of malicious IPs

i was testing somethings on browsers and i faced a case that when i browse meaningless addresses like `abc/`

`signortest/`

`word/`

the request sent to ip addresses hosted on linode! that some of them was reported malicious.

and i receive two different answers.(pictures attached)

- 4001
- 503 Service Unavailable

how i can understand what’s going on?

## geometry – Question about a series of distance preserving transformations on points

I have a problem that asks me to

Find all length preserving transformations of the plane that send

point A to point A’ and point B to point B’ where: $A=(0,1), B=(1,1), A’=(3,2), B’=(3- frac{sqrt3}{2}, frac{3}{2})$;

and to write the transformations as a parallel transport followed by a rotation about the origin, and possibly a reflection.

I did some preliminary work and found that for the rotation, $theta = frac{pi}{6}$. I’m now left with systems of equations that involve the variables of transport. Would it just remain to solve the system for those variables? And how can I determine if a reflection is needed? Do I need to take into account the possible reflection when I write the formula for the points after translation and rotation?