real analysis – Series of Product of Sequences

Studying for Exams, here’s the problem.

Let ${a_n}, {b_n}$ be two sequences in $mathbb{R}$ and set


Show the following

(a) $sum_{n=0}^{N}a_nb_n=sum_{n=0}^{N-1}A_n(b_n-b_{n+1})+A_Nb_N$

(b) If ${b_n}$ is decreasing and non-negative with $lim_{ntoinfty}b_n=0$, and if the sequence ${A_n}$ is bounded, then $sum_{n=0}^{infty}a_nb_n$ converges.

(c) If ${b_n}$ is decreasing and bounded below, and if $sum_{n=0}^{infty}a_n$ converges, then $sum_{n=0}^{infty}a_nb_n$ converges.

Here’s my thoughts so far.

(a) follows from induction.

(c) seems to follow from (b) if we let $c_n=b_n-lim_{ntoinfty}b$ (which exists since $b_n$) is monotone and bounded. If $sum_{n=0}^{infty}a_n$ converges, then the sequence of partial sums ${A_n}$ must be bounded, which gives the same conditions as (b).

I can’t get the solution for (b). We don’t have any information on non-negativity for ${a_n}$, so that shuts out a lot of routes for convergence. The form from (a) lends itself to check for some kind of telescoping but I haven’t made progress there. Is the Cauchy condition the way to go?

Any hints would be appreciated.

Product quick view

Would you add a product quick view option on hover in the product listing page? Maybe you have any experience with how users respond to such option?

Example can be seen here, where the "quick view" option appears on the product photo:

(it’s a test site)

rsa – One product unit needs one or two x.509 certificate

I’m developing a safety product and asked to enable secure communication support to work with maintenance application running on a host pc and to work with third-party products.

So Do I need to have two certificates in my product one with ECC key and another with RSA key so that the unit is capable of working with broader range of products out there in the market making use of ECC and RSA based certificates?

Also, tell me is it possible to establish a secure communication between two entities where one is using RSA cert and the other is using ECC cert? In my opinion, it is not possible as there is no such cipher but just wanted a confirmation.

As Issuing CA server has a limit on 20k certificates, would I not be reducing the capacity If I end up using two certificates? Are there any alternates for this?

theming – Get comments count in commerce product

I set up comments for a commerce product type called “produkt”. Now I want to get the comment count by preprocessing the product in my theme.theme file and get the comment count for the field “field_produktbewertungen”.

function theme_preprocess_commerce_product(&$variables) {
    $variables('ratings_count') = $variables('commerce_product')->get('field_produktbewertungen')->comment_count;

Is there anything different in preprocessing products instead of nodes?
Thank you very much!

magento2 – How to filter Order Collection by Specific Product Item name?

I would like to know how to filter order collection by using a product item name in magento 2?

$collection = $this->_orderCollectionFactory->create()->addAttributeToSelect('*');
        ("soi" => "sales_order_item"),
        'main_table.entity_id = soi.order_id',
     ->where('', array('like' => '% '.$params('filter-text').' %'))

Just tried this way and it didn’t work. I get no order collection using this collection filter. Any help will be appreciated! Thanks!

magento2.4 – magento 2 sort child products according swatches in product view page

I wanna sort the child products of configurable according to display swatches.
I wanna sort the child products in product page.

Product page swatches display like this Green,White,Black
When I use my below code its fetching child products but not sort according to swatches. How swatches sorted and display, and what should I change in my below code ?

$childItems = ();
        $_objectManager = MagentoFrameworkAppObjectManager::getInstance();
        $_product = $_objectManager->get('MagentoFrameworkRegistry')->registry('current_product');
            if($_product->getTypeId() == "configurable"){
            $imageBuilder= $_objectManager->get('MagentoCatalogModelProductImageUrlBuilder');
                $allProducts = $_product->getTypeInstance()->getUsedProducts($_product, null);
                foreach ($allProducts as $product) {
                    $productStockObj =  $objectManager->get('MagentoCatalogInventoryApiStockRegistryInterface')->getStockItem($product->getId());
                   if ($productStockObj->getIsInStock()){
                    $childItems() = $product->getSku();

When I run above code it displays the child array like this sku-white,sku-black,sku-green

I want sort the child items according display swatches

i will rank your website article and product on google and i will give you links list you can check for $10

i will rank your website article and product on google and i will give you links list you can check

Rank higher with relevant backlinks from a top-rated seller

This is for Manual Link Outreach, connecting your website with relevant sites using your anchor text and URLs.

This works by contacting industry-related sites. & opportunities are then explored to place links back to relevant pages on your site to help you to rank higher in the search results.

I target sites as close to your niche or industry as possible.

Why outreach?

Getting links as relevant as possible to your industry or niche will carry higher authority with the search engines. Getting a few links like this can produce more positive results than what you would get from thousands of very low-quality ones.

you will get links placed on sites with DA 10-30+ (usually higher. See PDF in gallery)

RD (see faq) will average 50. usually much higher.

I will do keyword research if needed & will advise on anchor text (keyword) use based on the current backlink profile.

Links are placed on real sites belonging to real people inside aged content that is already indexed

These are not from PBNs, networks, or guest posts dressed up as outreach.

Please see the FAQ below for answers to questions


magento2.3 – magento 2.3 observers on product attributes save/changes

on my magento 2.3 I need to detect when a particular product attribute is updated in order to change the product price. Changing the product attribute via backend is no big deal: I simply put an observer on the catalog_product_save_after event and check my current attribute value against $product->getOrigData('attribute_code') to see if it’s changed.

The problemi is: when the attribute is imported via csv import feature, that observer is not called. Instead, the catalog_product_import_bunch_save_after event is called. But here I have no way of retrieving the original value of the attribute (because it is called after the save) and the event catalog_product_import_bunch_save_before (that would hold the original data) doesn’t exist.

So now my question is: how can I get the original data for my attribute during the csv import? If I can’t with the observer I’m using, which observer/plugin should I use?


magento2 – Setting Up Custom URLs For Each Option/Swatch of a Variable Product

A client wanted to setup where he sells fabric and related stuff and wanted each variation on a different URL.

So example, he sells a shirt with the colors red, blue, and black. Upon landing on the product page, the URL would be: and upon clicking on a color, it would be (where the “red” after the base-t-shirt/ is the option SKU or whatever identifier)

I believe that this is not a native feature of Magento but is there any workaround?